Question

I'm totally confused on how to determine if a series is convergent or divergent.
So for this series: -20+100-500
I believe that it is convergent because r=-5.
However, I'm supposed to use the comparison test on this question:
(1/49)+(1/64)+(1/81)+...
And I have no idea how to go about it. Could you also help me understand how to find the general term of the series? Any help would be greatly appreciated!

Answer 1

The first series (assuming infinite series),
\[-20+100-500+\cdots=4\sum_{k=0}^\infty(-5)^{k+1}\]
diverges precisely because the ratio is \(r=-5\). A geometric series can only converge if \(|r|<1\).

Answer 2

For the second series, what you have is the sum of reciprocal squares,
\[\frac{1}{49}+\frac{1}{64}+\frac{1}{81}+\cdots=\frac{1}{7^2}+\frac{1}{8^2}+\frac{1}{9^2}+\cdots=\sum_{k=7}^\infty\frac{1}{k^2}\]
This series converges because it's smaller than another almost identical series,
\[\sum_{k=1}^\infty\frac{1}{k^2}=1+\frac{1}{4}+\frac{1}{9}+\cdots\]
(the index is different). The "comparison" series can be shown to converge using the integral test.

The reason series number 2 converges is because it has fewer terms, so it adds up to a number less than the comparison series' sum.