Test for convergence of the series sumn=1inftyleft(frac1n+1right)n. | Basic Mathematics 2080

Basic Mathematics20805 marksNumerical: series convergence testRoot test

Test for convergence of the series $\sum_{n=1}^{\infty}\left(\frac{1}{n+1}\right)^n$ . [5]

Test for convergence of the series $\sum_{n=1}^{\infty}\left(\frac{1}{n+1}\right)^n$. [5]

Series: $\displaystyle\sum{n=1}^{\infty}\left(\frac{1}{n+1}\right)^n$ - General term: $an = \left(\dfrac{1}{n+1}\right)^n$ - All terms positive, so ordinary convergence = absolute convergence. No data missing. For a positive-term series $\sum an$ , define $L = \lim{n\to\infty} \sqrt[n]{an}.$ -...

Series: $\displaystyle\sum{n=1}^{\infty}\left(\frac{1}{n+1}\right)^n$ - General term: $an = \left(\dfrac{1}{n+1}\right)^n$ - All terms positive, so ordinary convergence = absolute convergence. No data missing. For a positive-term series $\sum an$, define $$L = \lim{n\to\infty} \sqrt[n]{an}.$$ -...