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edited Feb 20, 2013 at 23:25

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In Calculus we teach that if the $a_n$ are positive and decreasing with limit equal to zero, then the alternating series $\sum_n (-1)^na_n$ converges. One can in general not leave out the assumption that the $a_n$ (eventually) decrease, as the example $a_{2n}:=1/n$ and $a_{2n+1}:=1/2^n$ shows. However, most examples are series where the $a_n$ are given by some function $a_n=f(n)$ (for $n\gg 0$).

So my question is, for which class of functions $\mathcal F$ do we have the property that if $f$ is a positive function in $\mathcal F$ and $\lim_{x\to \infty}f(x)=0$, then the alternating series $\sum_n (-1)^nf(n)$ converges. For instance, any o-minimal class will work, since any $f$ with limit zero at infinity must eventually be decreasing. But I think if we add the sine function to this class and close under addition, multiplication and composition, this is still true. In fact, I would almost dare to postulate that the class of elementary functions (i.e., the ones our students work with), have this property, and so we do not need to ``bug'' them with this extra condition.

In Calculus we teach that if the $a_n$ are positive and decreasing with limit equal to zero, then the alternating series $\sum_n (-1)^na_n$ converges. One can in general not leave out the assumption that the $a_n$ (eventually) decrease, as the example $a_{2n}:=1/n$ and $a_{2n+1}:=1/2^n$ shows. However, most examples are series where the $a_n$ are given by some function $a_n=f(n)$ (for $n\gg 0$). So my question is, for which class of functions $\mathcal F$ do we have the property that if $f$ is a positive function in $\mathcal F$ and $\lim_{x\to \infty}f(x)=0$, then the alternating series $\sum_n (-1)^nf(n)$ converges. For instance, any o-minimal class will work, since any $f$ with limit zero at infinity must eventually be decreasing. But I think if we add the sine function to this class and close under addition, multiplication and composition, this is still true. In fact, I would almost dare to postulate that the class of elementary functions (i.e., the ones our students work with), have this property, and so we do not need to ``bug'' them with this extra condition.

In Calculus we teach that if the $a_n$ are positive and decreasing with limit equal to zero, then the alternating series $\sum_n (-1)^na_n$ converges. One can in general not leave out the assumption that the $a_n$ (eventually) decrease, as the example $a_{2n}:=1/n$ and $a_{2n+1}:=1/2^n$ shows. However, most examples are series where the $a_n$ are given by some function $a_n=f(n)$ (for $n\gg 0$).

So my question is, for which class of functions $\mathcal F$ do we have the property that if $f$ is a positive function in $\mathcal F$ and $\lim_{x\to \infty}f(x)=0$, then the alternating series $\sum_n (-1)^nf(n)$ converges. For instance, any o-minimal class will work, since any $f$ with limit zero at infinity must eventually be decreasing. But I think if we add the sine function to this class and close under addition, multiplication and composition, this is still true. In fact, I would almost dare to postulate that the class of elementary functions (i.e., the ones our students work with), have this property, and so we do not need to ``bug'' them with this extra condition.

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asked Feb 20, 2013 at 23:14

558
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Alternating series test for non-decreasing terms

In Calculus we teach that if the $a_n$ are positive and decreasing with limit equal to zero, then the alternating series $\sum_n (-1)^na_n$ converges. One can in general not leave out the assumption that the $a_n$ (eventually) decrease, as the example $a_{2n}:=1/n$ and $a_{2n+1}:=1/2^n$ shows. However, most examples are series where the $a_n$ are given by some function $a_n=f(n)$ (for $n\gg 0$). So my question is, for which class of functions $\mathcal F$ do we have the property that if $f$ is a positive function in $\mathcal F$ and $\lim_{x\to \infty}f(x)=0$, then the alternating series $\sum_n (-1)^nf(n)$ converges. For instance, any o-minimal class will work, since any $f$ with limit zero at infinity must eventually be decreasing. But I think if we add the sine function to this class and close under addition, multiplication and composition, this is still true. In fact, I would almost dare to postulate that the class of elementary functions (i.e., the ones our students work with), have this property, and so we do not need to ``bug'' them with this extra condition.

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