MHB Help with understanding this series

[Linus12351]

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

 

[topsquark]

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

 

[Prove It]

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.