Timeline for Finding a variable P for which a sum converges [closed]
Current License: CC BY-SA 4.0
14 events
| when toggle format | what | by | license | comment | |
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| Jun 17, 2020 at 8:28 | history | closed |
user44191 Robert Israel Alex M. Ramiro de la Vega David Handelman |
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| Jun 15, 2020 at 15:09 | comment | added | Alapan Das | For , large $n$, $(1-\frac{1}{2n})≈1$, hence, it's equivalent to $\sum_{k=0} x^k$. | |
| Jun 15, 2020 at 15:03 | comment | added | Roee | @AlapanDas I don't know how to prove that $ \left(1-\frac{1}{n}\right)^p $ diverges | |
| Jun 15, 2020 at 14:57 | comment | added | Alapan Das | In your question, $\lim \limits_{n \to \infty} (1-\frac{1}{2n(1-\frac{1}{ln(n)})+1})^{\frac{-p}{n}}≈(1-\frac{1}{2n})^{\frac{-p}{n}}$. This becomes similar to asking whether $\sum_{n=0} (1-\frac{1}{2n})^p$ is convergent or not. And obviously this is divergent. | |
| Jun 15, 2020 at 14:46 | comment | added | Alapan Das | Yes you are right. Take the two series for example. 1) $f(x)=\sum_{n=1}^{\infty} \frac{x^n}{n}$ and 2)$g(x)=\sum_{n=1}^{\infty} \frac{x^n}{n^2}$. For both the series $\lim \limits_{n \to \infty} \sqrt[n]{|a_n|}=1$. But, at $x=1$, $f(x)$ doesn't converge, but $g(x)$ does. | |
| Jun 15, 2020 at 14:39 | comment | added | Roee | @AlapanDas I've been taught that if $\lim _{n\to \infty \:}\left(\sqrt[n]{a_n}\right)=1$ the root test is indecisive, can you please explain further? | |
| Jun 15, 2020 at 13:52 | comment | added | Dieter Kadelka | To show that $\Sigma (1-(1/n))^p = \infty$ assume that $p \in \mathbb{N}$ and use the $\zeta$-function. | |
| Jun 15, 2020 at 13:42 | review | Close votes | |||
| Jun 17, 2020 at 8:28 | |||||
| Jun 15, 2020 at 13:32 | comment | added | Roee | @GabeConant I've tried using the direct comprasion test on $a_n$ with the sequence $ (1-(1/n))^p $ which I know diverges, but I'm not sure how to prove it. | |
| Jun 15, 2020 at 13:28 | comment | added | Roee | @DieterKadelka I've tried using the Stirling formula, but I didn't see where it can lead. | |
| Jun 15, 2020 at 13:16 | comment | added | Gabe Conant | Have you tried the divergence test on $a_n$? | |
| Jun 15, 2020 at 13:08 | comment | added | Dieter Kadelka | Have you tried replacing $\ln(n!)$ with the Stirling formula? | |
| Jun 15, 2020 at 12:56 | review | First posts | |||
| Jun 15, 2020 at 12:59 | |||||
| Jun 15, 2020 at 12:49 | history | asked | Roee | CC BY-SA 4.0 |