Help with understanding this series

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The discussion focuses on the convergence of the series $\sum{\frac{1}{5^{n-1} + 1}}$. Dan suggests using the limit comparison test, specifically $\lim_{n \to \infty} \dfrac{a_{n + 1}}{a_n}$, to analyze the series. Another participant demonstrates that the series converges by comparing it to a convergent geometric series, concluding that since the positive term series is less than a known convergent series, it also converges by the comparison test.

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Can anyone help with this problem. I've tried integral test but seems to be too complicated.View attachment 9622
 

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Linus12351 said:
Can anyone help with this problem. I've tried integral test but seems to be too complicated.
Have you tried looking at [math]\lim_{n \to \infty} \dfrac{a_{n + 1}}{a_n}[/math]?

-Dan
 
Linus12351 said:
Can anyone help with this problem. I've tried integral test but seems to be too complicated.

Easy,

$\displaystyle 0 \leq \sum{\frac{1}{5^{n-1} + 1}} < \sum{\frac{1}{5^{n-1}}} = \sum{ \left( \frac{1}{5} \right) ^{n-1} }$

Since your positive term series is less than a convergent geometric series, your series converges by comparison.
 

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