find a formula for the nth partial sum of the series and use it to determine if the series converges or…
find a formula for the nth partial sum of the series and use it to determine if the series converges or diverges. if the series converges, find its sum\n sum _ { n = 1 } ^ { infty } left( \frac { 12 } { n } - \frac { 12 } { n + 1 } \right) \n( s _ { n } = )
Answer
Explanation:
Step1: Write out the partial sum formula
The (n)th partial sum (S_n=\sum_{k = 1}^{n}\left(\frac{12}{k}-\frac{12}{k + 1}\right)). Expand the sum: [ \begin{align*} S_n&=12\left[\left(1-\frac{1}{2}\right)+\left(\frac{1}{2}-\frac{1}{3}\right)+\cdots+\left(\frac{1}{n}-\frac{1}{n + 1}\right)\right]\ \end{align*} ]
Step2: Simplify the sum
By the telescoping - series property (most terms cancel out). (S_n = 12\left(1-\frac{1}{n+1}\right)=\frac{12n}{n + 1})
Step3: Determine convergence
Find (\lim_{n\rightarrow\infty}S_n). [ \begin{align*} \lim_{n\rightarrow\infty}S_n&=\lim_{n\rightarrow\infty}\frac{12n}{n + 1}\ &=\lim_{n\rightarrow\infty}\frac{12}{1+\frac{1}{n}}\ \end{align*} ] As (n\rightarrow\infty), (\frac{1}{n}\rightarrow0).
Answer:
The (n)th partial sum (S_n=\frac{12n}{n + 1}). The series converges and its sum is (12).