Determine whether each of the following series is convergent or divergent.
(a) \(\displaystyle{\sum_{n=1}^{\infty} \frac{(-2)^n}{n}}\)
solution I
solution II
(b) \(\displaystyle{\sum_{n=1}^{\infty} \frac{(n+1)\,3^n}{2^n\,n^3}}\) solution
(c) \(\displaystyle{\sum_{k=1}^{\infty} \frac{3}{k!}}\) solution
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(d) \(\displaystyle{\sum_{n=1}^{\infty} \left( \frac{n^2+1}{3n^2+2}\right)^n}\) solution
(e) \(\displaystyle{\sum_{m=1}^{\infty} \frac{m}{e^m}}\) solution
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(f) \(\displaystyle{\sum_{n=1}^{\infty} \left( \frac{5n^2-2}{4n^2+3}\right)^n}\) solution
(g) \(\displaystyle{\sum_{n=1}^{\infty} \frac{n!}{n^n}}\) solution
(h) \(\displaystyle{\sum_{n=1}^{\infty}(-1)^{n-1} \:
\frac{7^{n}}{5^n\, n^3}}\) solution
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(i) \(\displaystyle{\sum_{n=3}^{\infty} 4 \left(1+ \frac{1}{n}\right)^{n^2}}\) solution
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