Question

Find the exact sum of the series

S = ∑ (5^(n+2))/(2^(n)*pi^(n+1)) for n = 1 to infinity

Answer 1

looking at the terms

\[T_{1} = \frac{5^3}{2\pi^2}\]

\[T_{2} = \frac{5^4}{2^2\pi^3}\]

so it would appear the common ratio

is

\[r = \frac{5}{2\pi}\]

since the common ratio is

-1 < r < 1

it means the series will have a limiting sum

\[S_{\infty} = \frac{a}{1 - r}\]

a is the 1st term, shown above and r is the common ratio.

Hope this helps.

Answer 2

thank you!