Question

3/(n(n+3)) convergent or divergent?
i know its telescoping but when i do partial fratcion decomp im not seeing a pattern and im not sure where i went wrong

Answer 1

im made (A/n)+(B/(n+3)) and after going all the way thur i found that A+B=0 and 3A=3 thus A=1 and B=-1 when pluged back in i do not see a pattern when i start to plug in the numbers to see the sequence, im not able to cancel anything.

Answer 2

\[\frac{1}{n}-\frac{1}{n+3}\] yes, your partial sum is correct :)

Now, I am assuming this is a sum from n=1 to infinity? i.e. \[ \sum_{n=1}^{\infty}\left(\frac{1}{n}-\frac{1}{n+3}\right)\]

Let \(S_n\) be the partial sum
so:
\[S_n=\left(\frac{1}{1}-\frac{1}{1+3}\right)+\left(\frac{1}{2}-\frac{1}{2+3}\right)+\left(\frac{1}{3}-\frac{1}{3+3}\right)+\left(\frac{1}{4}-\frac{1}{4+3}\right)+\ldots\\
\, \,\,+\left(\frac{1}{n-3}-\frac{1}{n}\right)+\left(\frac{1}{n-2}-\frac{1}{n+1}\right)+\left(\frac{1}{n-1}-\frac{1}{n+2}+\right)+\left(\frac{1}{n}-\frac{1}{n+3}\right)\\
\,\\
\,\\
\, \\

S_n=\left(1-\cancel{\frac{1}{4}}\right)+\left(\frac{1}{2}-\cancel{\frac{1}{5}}\right)+\left(\frac{1}{3}-\cancel{\frac{1}{6}}\right)+\left(\cancel{\frac{1}{4}}-\cancel{\frac{1}{7}}\right)+\left(\cancel{\frac{1}{5}}-\cancel{\frac{1}{8}}\right)+\ldots\\
\,\,+\left(\cancel{\frac{1}{n-3}}-\cancel{\frac{1}{n}}\right)+\left(\cancel{\frac{1}{n-2}}-\frac{1}{n+1}\right)+\left(\cancel{\frac{1}{n-1}}-\frac{1}{n+2}\right)+\left(\cancel{\frac{1}{n}}-\frac{1}{n+3}\right)\\
\,\, \\
\, \\

S_n=1+\frac{1}{2}+\frac{1}{3}-\frac{1}{n+1}-\frac{1}{n+2}-\frac{1}{n+3}\]

Then, you take the limit of the partial sum
\[ \lim_{n\rightarrow\infty}S_n=\lim_{n\rightarrow\infty}\left( 1+\frac{1}{2}+\frac{1}{3}-\frac{1}{n+1}-\frac{1}{n+2}-\frac{1}{n+3}\right)=1+\frac{1}{2}+\frac{1}{3}=\frac{11}{6}\]
Hence, it is convergent.

Answer 3

Hm in the second \(S_n\), on the second line, it it just the same \(\large \frac{1}{n}-\frac{1}{n-3}\) term written but got cut off.

Answer 4

thank you so much! i was right on track, you are amazing!!!! thank you for taking the time to help me with that!!!!

Answer 5

:) yw