Consider the series $\sum^\infty_{n=1} \frac {2}{n^2+2n}$.
c) According to the integral test, how many terms should we add to ensure
that the difference between S and and the partial sum $S_k$is not greater than 1/100?
I started by using the Remainder Theorem for the Integral Test.
The theorem says suppose that $f(k)=a_k$ where f is a continuous, positive, decreasing function for $x \ge n$ and $\sum a_n$ is convergent. If $R_n =s-s_n$, then
$\int^\infty_{n+1} f(x) dx \le R_n \le \int^\infty_{n} f(x) dx$.
So I let $f(x)=\frac {2}{n^2+2n}$. I know that the function is convergent, continuous and decreasing, I don't know how to prove that it's positive.
I first started by finding $\int^\infty_{n} f(x) dx$ since the question says $R_n$ not greater than 1/100.
I found that $\int^\infty_{n} \frac {2}{n^2+2n} dx= ln \mid \frac {n+2}{n} \mid$.
With this I found that $R_n \le \int^\infty_{n} f(x) dx<1/100$.
This is where I am stuck. I tried,
$ln \mid \frac {n+2}{n} \mid<1/100$
$e^{ln \mid \frac {n+2}{n} \mid}<e^{1/100}$
$\mid \frac {n+2}{n} \mid< e^{1/100}$.
I don't know what to do after this.