# Thread: Infinite Sequences & Series Question

1. ## Infinite Sequences & Series Question

Have a few pressing questions:

1) Test for Convergence or Divergence; if Convergent find sum. Sum (n = 1 to Infinity) of 10^n/(-9)^(n-1). Can I make this (10^(n-1)*10)/(-9)^(n-1) = -10/9^(n-1)*10? The -9 is in parenthesis in the original problem, and I'm not sure if I can do that. It would diverge as r = -10/9, |-10/9| = 10/9 > 1, but again not sure.

2) Determine covergence or divergence by expressing s(subn) as a telescoping sum. If convergent, find sum:

Sum (n = 1 to Inifinity) of 2/(n^2 + 4n + 3). I see the factor (n + 1)(n+3), but aren't familiar enough with Telescoping series to know how to proceed.

Thanks!

2. Sum (n = 1 to Inifinity) of 2/(n^2 + 4n + 3). I see the factor (n + 1)(n+3), but aren't familiar enough with Telescoping series to know how to proceed.

Thanks!
Expand and get $\displaystyle \frac{1}{n+1}-\frac{1}{n+3}$

$\displaystyle \left(\frac{1}{2}-\frac{1}{4}\right)+\left(\frac{1}{3}-\frac{1}{5}\right)+\left(\frac{1}{4}-\frac{1}{6}\right)+\left(\frac{1}{5}-\frac{1}{7}\right)+................$

as we can see, all the terms cancel one another out except for $\displaystyle \frac{1}{2}+\frac{1}{3}=\frac{5}{6}$. See what is meant by telescoping now?.