(a) \(\displaystyle{\sum_{n=1}^{\infty} \frac{\sin n}{n^2}}\)
solution
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(b) \(\displaystyle{\sum_{n=1}^{\infty} (-1)^{n-1} \: \frac{1}{2n+3}}\)
solution
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(c) \(\displaystyle{\sum_{n=1}^{\infty}
(-1)^{n-1} \: \frac{1}{n}}\)
solution
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(d) \(\displaystyle{\sum_{n=1}^{\infty} (-1)^{n+1} \: \frac{2^n-1}{3^n-1}}\)
solution
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(e) \( \displaystyle{\sum_{n=1}^{\infty}
\frac{(-2)^n}{(2n+1)!}}\) solution
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(f) \(\displaystyle{\sum_{n=1}^{\infty}
\frac{n!}{{100}^n}}\) solution
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(g) \( \displaystyle{\sum_{n=1}^{\infty}
(-1)^n\frac{1}{\sqrt[3]{n}}}\) solution
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(h) \(\displaystyle{\sum_{n=1}^{\infty}
(-1)^{n-1}\,\frac{2n+5}{3n+1}}\) solution
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(a) \(\displaystyle{\sum_{n=1}^{\infty} \frac{(-2)^n}{n}}\)
sol I
sol II
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(b) \(\displaystyle{\sum_{n=1}^{\infty} \frac{(n+1)\,3^n}{2^n\,n^3}}\) solution
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(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
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(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
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(g) \(\displaystyle{\sum_{n=1}^{\infty} \frac{n!}{n^n}}\) solution
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(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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