Question

Need Calc Help!!! Please!
For each series given, state convergent or divergent. If convergent, find the sum. Justify all answers mathematically.

(4^n+1)/(5^n) for n=0 to infinity

Answer 1

@satellite73

Answer 2

\[\begin{align*}\sum_{n=0}^\infty \frac{4^n+1}{5^n}&=\sum_{n=0}^\infty \frac{4^n}{5^n}+\sum_{n=0}^\infty \frac{1}{5^n}\\
&=\sum_{n=0}^\infty \left(\frac{4}{5}\right)^n+\sum_{n=0}^\infty \left(\frac{1}{5}\right)^n\end{align*}\]
What do you know about geometric series and when they converge/diverge?

Answer 3

not really general idea but i am not confident with it yet

Answer 4

a geometric series \[\Large \sum_{n=0}^\infty r^n\] is convergent if \[\Large -1<r<1\] and divergent otherwise

Answer 5

furthermore, if the series is convergent, then
\[\Large \sum_{n=0}^{\infty} r^n = \frac{1}{1-r}\]

Answer 6

Thank you, which series has to do with limits and whether they exist or not