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To help avoid any misunderstanding that may arise for readers of this question, let me say that when understood in the usual sense, there are no nontrivial convergent sequences or series at all in the hyperreal numbers (and neither in the surreal numbers).

The reason is that the hyperreal numbers are usually understood to be countably saturated. This has a model-theoretic definition, meaning that every finitely realized type with countably many parameters is realized. But in the case of ordered fields, it can be expressed as the following property: whenever $$x_0\leq x_1\leq x_2\leq\cdots y_2\leq y_1\leq y_0$$ with $x_n<y_n$ for all $n\in\mathbb{N}$, then there is a number $z$ in between $$x_0\leq x_1\leq x_2\leq\cdots z \cdots y_2\leq y_1\leq y_0.$$

This saturation property prevents any nontrivial convergent sequence $x_n\to x$, unless it is eventually constant, since there will be various numbers $z$ in between, on each side.

Indeed, every countable set of hyperreal numbers is discrete in the hyperreal order. For this reason, one cannot apply any of the usual treatment of sequences and series from the real numbers in the hyperreal field.

Meanwhile, there is nevertheless a robust theory of sequences and series in the hyperreals, but using sequences and series indexed instead by $\mathbb{N}^*$, the nonstandard natural numbers, instead of merely $\mathbb{N}$. With this change, one can form hyperreal analogues of all the familiar sequences and series in the reals. The reason is that by the transfer principle, every assertion made in the real numbers about any such sequence or series is also true exactly the same in the nonstandard realm.

Thus, the radius of convergent of the nonstandard analogue of a power series remains the same as what it was.

For example, the series $$\sum_{n\in\mathbb{N}}\frac 1{2^n}=\frac12+\frac14+\cdots+\frac1{2^n}+\cdots$$ does not converge in the hyperreals, since one is using the natural numbers only. But the much longer (uncountably so) series $$\sum_{n\in\mathbb{N}^*}\frac 1{2^n}=\frac12+\frac14+\cdots+\frac1{2^n}+\cdots+\cdots=1$$ does converge to $1$.

In my essay on the surreal numbers, an elementary introduction, I mentioned this situation as a possible red flag for surreal calculus. To do calculus with the surreal numbers (or the hyperreals), one simply cannot expect to use sequences and series and convergence ideas in the same manner that one does with the real numbers. Rather, one must in a sense remount calculus entirely bottom-up on the new setting, using infinitesimals, saturation, and transfer.

Some accounts of nonstandard analysis seem to provide a way to avoid the issue by simply treating $\mathbb{N}^*$ as the genuine natural numbers, and then talking some the standard numbers as a special kind of number. In this way of speaking, the subject of nonstandard analysis is concerned with sequences and series, even though the index set is uncountable by the usual way of talking. In my experience, however, this practice often leads to confusion when mathematicians from other areas join in. And so I find it best to be clear about what one means.

(Lastly, let me add that unless one adopts additional axioms, it is not usually correct to refer to "the" hyperreals, since we have no categorical such structure. But see my recent paper How CH could have been a fundamental axiom for more about how it could have been different.)

To help avoid any misunderstanding that may arise for readers of this question, let me say that when understood in the usual sense, there are no nontrivial convergent sequences or series at all in the hyperreal numbers (and neither in the surreal numbers).

The reason is that the hyperreal numbers are usually understood to be countably saturated. This has a model-theoretic definition, meaning that every finitely realized type with countably many parameters is realized. But in the case of ordered fields, it can be expressed as the following property: whenever $$x_0\leq x_1\leq x_2\leq\cdots \leq y_2\leq y_1\leq y_0$$ with $x_n<y_n$ for all $n\in\mathbb{N}$, then there is a number $z$ in between $$x_0\leq x_1\leq x_2\leq\cdots <z< \cdots \leq y_2\leq y_1\leq y_0.$$

This saturation property prevents any nontrivial convergent sequence $x_n\to x$, unless it is eventually constant, since there will be various numbers $z$ in between, on each side.

Indeed, every countable set of hyperreal numbers is discrete in the hyperreal order. For this reason, one cannot apply any of the usual treatment of sequences and series from the real numbers in the hyperreal field.

Meanwhile, there is nevertheless a robust theory of sequences and series in the hyperreals, but using sequences and series indexed instead by $\mathbb{N}^*$, the nonstandard natural numbers, instead of merely $\mathbb{N}$. With this change, one can form hyperreal analogues of all the familiar sequences and series in the reals. The reason is that by the transfer principle, every assertion made in the real numbers about any such sequence or series is also true exactly the same in the nonstandard realm.

Thus, the radius of convergent of the nonstandard analogue of a power series remains the same as what it was.

For example, the series $$\sum_{n\in\mathbb{N}}\frac 1{2^n}=\frac12+\frac14+\cdots+\frac1{2^n}+\cdots$$ does not converge in the hyperreals, since one is using the natural numbers only. But the much longer (uncountably so) series $$\sum_{n\in\mathbb{N}^*}\frac 1{2^n}=\frac12+\frac14+\cdots+\frac1{2^n}+\cdots+\cdots=1$$ does converge to $1$.

In my essay on the surreal numbers, an elementary introduction, I mentioned this situation as a possible red flag for surreal calculus. To do calculus with the surreal numbers (or the hyperreals), one simply cannot expect to use sequences and series and convergence ideas in the same manner that one does with the real numbers. Rather, one must in a sense remount calculus entirely bottom-up on the new setting, using infinitesimals, saturation, and transfer.

Some accounts of nonstandard analysis seem to provide a way to avoid the issue by simply treating $\mathbb{N}^*$ as the genuine natural numbers, and then talking some the standard numbers as a special kind of number. In this way of speaking, the subject of nonstandard analysis is concerned with sequences and series, even though the index set is uncountable by the usual way of talking. In my experience, however, this practice often leads to confusion when mathematicians from other areas join in. And so I find it best to be clear about what one means.

(Lastly, let me add that unless one adopts additional axioms, it is not usually correct to refer to "the" hyperreals, since we have no categorical such structure. But see my recent paper How CH could have been a fundamental axiom for more about how it could have been different.)

To help avoid any misunderstanding that may arise for readers of this question, let me say that when understood in the usual sense, there are no nontrivial convergent sequences or series at all in the hyperreal numbers (and neither in the surreal numbers).

The reason is that the hyperreal numbers are usually understood to be countably saturated. This has a model-theoretic definition, meaning that every finitely realized type with countably many parameters is realized. But in the case of ordered fields, it can be expressed as the following property: whenever $$x_0\leq x_1\leq x_2\leq\cdots y_2\leq y_1\leq y_0$$ with $x_n<y_n$ for all $n\in\mathbb{N}$, then there is a number $z$ in between $$x_0\leq x_1\leq x_2\leq\cdots z \cdots y_2\leq y_1\leq y_0.$$

This saturation property prevents any nontrivial convergent sequence $x_n\to x$, unless it is eventually constant, since there will be various numbers $z$ in between, on each side.

Indeed, every countable set of hyperreal numbers is discrete in the hyperreal order. For this reason, one cannot apply any of the usual treatment of sequences and series from the real numbers in the hyperreal field.

Meanwhile, there is nevertheless a robust theory of sequences and series in the hyperreals, but using sequences and series indexed instead by $\mathbb{N}^*$, the nonstandard natural numbers, instead of merely $\mathbb{N}$. With this change, one can form hyperreal analogues of all the familiar sequences and series in the reals. The reason is that by the transfer principle, every assertion made in the real numbers about any such sequence or series is also true exactly the same in the nonstandard realm.

Thus, the radius of convergent of the nonstandard analogue of a power series remains the same as what it was.

For example, the series $$\sum_{n\in\mathbb{N}}\frac 1{2^n}=\frac12+\frac14+\cdots+\frac1{2^n}+\cdots$$ does not converge in the hyperreals, since one is using the natural numbers only. But the much longer (uncountably so) series $$\sum_{n\in\mathbb{N}^*}\frac 1{2^n}=\frac12+\frac14+\cdots+\frac1{2^n}+\cdots+\cdots=1$$ does converge to $1$.

In my essay on the surreal numbers, an elementary introduction, I mentioned this situation as a possible red flag for surreal calculus. To do calculus with the surreal numbers (or the hyperreals), one simply cannot expect to use sequences and series and convergence ideas in the same manner that one does with the real numbers. Rather, one must in a sense remount calculus entirely bottom-up on the new setting, using infinitesimals, saturation, and transfer.

Sometimes hardcore nonstandard analysis people try to mount a version of nonstandard analysis that seeks to avoid the pain by simply treating $\mathbb{N}^*$ as the real natural numbers, and then talking sometimes about standard numbers as a special kind of number. And so they take themselves to be talking about sequences and series, even though their index set is uncountable by the usual way of talking. In my view, the mathematicians who use this vocabularly often have a hard time communicating with other mathematicians who have not yet bought in with that enormous change. And so my recommendation is for clarity in presentation about what one is talking about.

(Lastly, let me add that unless one adopts additional axioms, it is not usually correct to refer to "the" hyperreals, since we have no categorical such structure. But see my recent paper How CH could have been a fundamental axiom for more about how it could have been different.)

To help avoid any misunderstanding that may arise for readers of this question, let me say that when understood in the usual sense, there are no nontrivial convergent sequences or series at all in the hyperreal numbers (and neither in the surreal numbers).

The reason is that the hyperreal numbers are usually understood to be countably saturated. This has a model-theoretic definition, meaning that every finitely realized type with countably many parameters is realized. But in the case of ordered fields, it can be expressed as the following property: whenever $$x_0\leq x_1\leq x_2\leq\cdots y_2\leq y_1\leq y_0$$ with $x_n<y_n$ for all $n\in\mathbb{N}$, then there is a number $z$ in between $$x_0\leq x_1\leq x_2\leq\cdots z \cdots y_2\leq y_1\leq y_0.$$

This saturation property prevents any nontrivial convergent sequence $x_n\to x$, unless it is eventually constant, since there will be various numbers $z$ in between, on each side.

Indeed, every countable set of hyperreal numbers is discrete in the hyperreal order. For this reason, one cannot apply any of the usual treatment of sequences and series from the real numbers in the hyperreal field.

Meanwhile, there is nevertheless a robust theory of sequences and series in the hyperreals, but using sequences and series indexed instead by $\mathbb{N}^*$, the nonstandard natural numbers, instead of merely $\mathbb{N}$. With this change, one can form hyperreal analogues of all the familiar sequences and series in the reals. The reason is that by the transfer principle, every assertion made in the real numbers about any such sequence or series is also true exactly the same in the nonstandard realm.

Thus, the radius of convergent of the nonstandard analogue of a power series remains the same as what it was.

For example, the series $$\sum_{n\in\mathbb{N}}\frac 1{2^n}=\frac12+\frac14+\cdots+\frac1{2^n}+\cdots$$ does not converge in the hyperreals, since one is using the natural numbers only. But the much longer (uncountably so) series $$\sum_{n\in\mathbb{N}^*}\frac 1{2^n}=\frac12+\frac14+\cdots+\frac1{2^n}+\cdots+\cdots=1$$ does converge to $1$.

In my essay on the surreal numbers, an elementary introduction, I mentioned this situation as a possible red flag for surreal calculus. To do calculus with the surreal numbers (or the hyperreals), one simply cannot expect to use sequences and series and convergence ideas in the same manner that one does with the real numbers. Rather, one must in a sense remount calculus entirely bottom-up on the new setting, using infinitesimals, saturation, and transfer.

Some accounts of nonstandard analysis seem to provide a way to avoid the issue by simply treating $\mathbb{N}^*$ as the genuine natural numbers, and then talking some the standard numbers as a special kind of number. In this way of speaking, the subject of nonstandard analysis is concerned with sequences and series, even though the index set is uncountable by the usual way of talking. In my experience, however, this practice often leads to confusion when mathematicians from other areas join in. And so I find it best to be clear about what one means.

(Lastly, let me add that unless one adopts additional axioms, it is not usually correct to refer to "the" hyperreals, since we have no categorical such structure. But see my recent paper How CH could have been a fundamental axiom for more about how it could have been different.)

To help avoid any misunderstanding that may arise for readers of this question, let me say that when understood in the usual sense, there are no nontrivial convergent sequences or series at all in the hyperreal numbers (and neither in the surreal numbers).

The reason is that the hyperreal numbers are usually understood to be countably saturated. This has a model-theoretic definition, meaning that every finitely realized type with countably many parameters is realized. But in the case of ordered fields, it can be expressed as the following property: whenever $$x_0\leq x_1\leq x_2\leq\cdots y_2\leq y_1\leq y_0$$ with $x_n<y_n$ for all $n\in\mathbb{N}$, then there is a number $z$ in between $$x_0\leq x_1\leq x_2\leq\cdots z \cdots y_2\leq y_1\leq y_0.$$

This saturation property prevents any nontrivial convergent sequence $x_n\to x$, unless it is eventually constant, since there will be various numbers $z$ in between, on each side.

Indeed, every countable set of hyperreal numbers is discrete in the hyperreal order. For this reason, one cannot apply any of the usual treatment of sequences and series from the real numbers in the hyperreal field.

Meanwhile, there is nevertheless a robust theory of sequences and series in the hyperreals, but using sequences and series indexed instead by $\mathbb{N}^*$, the nonstandard natural numbers, instead of merely $\mathbb{N}$. With this change, one can form hyperreal analogues of all the familiar sequences and series in the reals. The reason is that by the transfer principle, every assertion made in the real numbers about any such sequence or series is also true exactly the same in the nonstandard realm.

Thus, the radius of convergent of the nonstandard analogue of a power series remains the same as what it was.

For example, the series $$\sum_{n\in\mathbb{N}}\frac 1{2^n}=\frac12+\frac14+\cdots+\frac1{2^n}+\cdots$$ does not converge in the hyperreals, since one is using the natural numbers only. But the much longer (uncountably so) series $$\sum_{n\in\mathbb{N}^*}\frac 1{2^n}=\frac12+\frac14+\cdots+\frac1{2^n}+\cdots+\cdots=1$$ does converge to $1$.

In my essay on the surreal numbers, an elementary introduction, I mentioned this situation as a possible red flag for surreal calculus. To do calculus with the surreal numbers (or the hyperreals), one simply cannot expect to use sequences and series and convergence ideas in the same manner that one does with the real numbers. Rather, one must in a sense remount calculus entirely bottom-up on the new setting, using infinitesimals, saturation, and transfer.

Sometimes hardcore standard analysis people try to mount a version of nonstandard analysis that seeks to avoid the pain by simply treating $\mathbb{N}^*$ as the real natural numbers, and then talking sometimes about standard numbers as a special kind of number. And so they take themselves to be talking about sequences and series, even though their index set is uncountable by the usual way of talking. In my view, the mathematicians who use this vocabularly often have a hard time communicating with other mathematicians who have not yet bought in with that enormous change. And so my recommendation is for clarity in presentation about what one is talking about.

(Lastly, let me add that unless one adopts additional axioms, it is not usually correct to refer to "the" hyperreals, since we have no categorical such structure. But see my recent paper How CH could have been a fundamental axiom for more about how it could have been different.)

To help avoid any misunderstanding that may arise for readers of this question, let me say that when understood in the usual sense, there are no nontrivial convergent sequences or series at all in the hyperreal numbers (and neither in the surreal numbers).

The reason is that the hyperreal numbers are usually understood to be countably saturated. This has a model-theoretic definition, meaning that every finitely realized type with countably many parameters is realized. But in the case of ordered fields, it can be expressed as the following property: whenever $$x_0\leq x_1\leq x_2\leq\cdots y_2\leq y_1\leq y_0$$ with $x_n<y_n$ for all $n\in\mathbb{N}$, then there is a number $z$ in between $$x_0\leq x_1\leq x_2\leq\cdots z \cdots y_2\leq y_1\leq y_0.$$

This saturation property prevents any nontrivial convergent sequence $x_n\to x$, unless it is eventually constant, since there will be various numbers $z$ in between, on each side.

Indeed, every countable set of hyperreal numbers is discrete in the hyperreal order. For this reason, one cannot apply any of the usual treatment of sequences and series from the real numbers in the hyperreal field.

Meanwhile, there is nevertheless a robust theory of sequences and series in the hyperreals, but using sequences and series indexed instead by $\mathbb{N}^*$, the nonstandard natural numbers, instead of merely $\mathbb{N}$. With this change, one can form hyperreal analogues of all the familiar sequences and series in the reals. The reason is that by the transfer principle, every assertion made in the real numbers about any such sequence or series is also true exactly the same in the nonstandard realm.

Thus, the radius of convergent of the nonstandard analogue of a power series remains the same as what it was.

For example, the series $$\sum_{n\in\mathbb{N}}\frac 1{2^n}=\frac12+\frac14+\cdots+\frac1{2^n}+\cdots$$ does not converge in the hyperreals, since one is using the natural numbers only. But the much longer (uncountably so) series $$\sum_{n\in\mathbb{N}^*}\frac 1{2^n}=\frac12+\frac14+\cdots+\frac1{2^n}+\cdots+\cdots=1$$ does converge to $1$.

In my essay on the surreal numbers, an elementary introduction, I mentioned this situation as a possible red flag for surreal calculus. To do calculus with the surreal numbers (or the hyperreals), one simply cannot expect to use sequences and series and convergence ideas in the same manner that one does with the real numbers. Rather, one must in a sense remount calculus entirely bottom-up on the new setting, using infinitesimals, saturation, and transfer.

Sometimes hardcore nonstandard analysis people try to mount a version of nonstandard analysis that seeks to avoid the pain by simply treating $\mathbb{N}^*$ as the real natural numbers, and then talking sometimes about standard numbers as a special kind of number. And so they take themselves to be talking about sequences and series, even though their index set is uncountable by the usual way of talking. In my view, the mathematicians who use this vocabularly often have a hard time communicating with other mathematicians who have not yet bought in with that enormous change. And so my recommendation is for clarity in presentation about what one is talking about.

(Lastly, let me add that unless one adopts additional axioms, it is not usually correct to refer to "the" hyperreals, since we have no categorical such structure. But see my recent paper How CH could have been a fundamental axiom for more about how it could have been different.)