Question

In the following problem, determine if the series converges. If it does, is the convergence
conditional or absolute?

a) \[\sum_{n=1}^{\infty} (3+4^{n})/5^{n} \]

Answer 1

\[\sum_{n=1}^{\infty} (3+4^{n})/5^{n} \]

Answer 2

I would break the sum apart:
\[\sum_{1}^{\infty}\frac{3+4^n}{5^n} = \sum_{1}^{\infty}\frac{3}{5^n}+\frac{4^n}{5^n} = \sum_{1}^{\infty}\frac{3}{5^n}+\sum_{1}^{\infty}\Big(\frac{4}{5}\Big)^n\]
Its easy to see that both summations represent geometric series, which both have a common ratio less than one, so they each converge individually.
As such, their sum also converges.

Answer 3

What would the value of 'a' and 'r' be in the case of \[3/5^{n} \]

Answer 4

the first term is when n = 1, so that would be 3/5. Each term after that is calculated by multiplying by 1/5. Write the first few terms of the series to see it better:
\[\sum_{n=1}^{\infty}\frac{3}{5^n} = \frac{3}{5}+\frac{3}{25}+\frac{3}{125}+\frac{3}{625}+\cdots \]