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Exercise: Which of the following series are convergent? Justify your answers. You can use any convergence tests studied in Section 2.7 and those on the other page of this handout. (a) $\sum_{n=1}^{\infty} \frac{1}{(n^2+1)^{1/3}}$ (b) $\sum_{n=1}^{\infty} \sin(\frac{1}{n})$ (c) $\sum_{n=1}^{\infty} \frac{\log n}{n}$ (d) $\sum_{n=1}^{\infty} \frac{n^2}{n!}$ (e) $\sum_{n=1}^{\infty} \frac{2^n n!}{n^n}$ (f) $\sum_{n=1}^{\infty} \frac{1}{(\log n)^k}$ for some integer $k$. (g) $\sum_{n=1}^{\infty} \frac{1}{n(\log n)^2}$ (h) $\sum_{n=1}^{\infty} \frac{1}{(\log n)^n}$ (i) $\sum_{n=1}^{\infty} \frac{(-1)^n}{(\log n)^n}$

          Exercise: Which of the following series are convergent? Justify your answers. You can use any convergence tests studied in Section 2.7 and those on the other page of this handout.
(a) $\sum_{n=1}^{\infty} \frac{1}{(n^2+1)^{1/3}}$
(b) $\sum_{n=1}^{\infty} \sin(\frac{1}{n})$
(c) $\sum_{n=1}^{\infty} \frac{\log n}{n}$
(d) $\sum_{n=1}^{\infty} \frac{n^2}{n!}$
(e) $\sum_{n=1}^{\infty} \frac{2^n n!}{n^n}$
(f) $\sum_{n=1}^{\infty} \frac{1}{(\log n)^k}$ for some integer $k$.
(g) $\sum_{n=1}^{\infty} \frac{1}{n(\log n)^2}$
(h) $\sum_{n=1}^{\infty} \frac{1}{(\log n)^n}$
(i) $\sum_{n=1}^{\infty} \frac{(-1)^n}{(\log n)^n}$

Exercise: Which of the following series are convergent? Justify your answers. You can use any convergence tests studied in Section 2.7 and those on the other page of this handout.
(a) ∑n=1^∞(1)/((n^2+1)^1/3)
(b) ∑n=1^∞sin((1)/(n))
(c) ∑n=1^∞(log n)/(n)
(d) ∑n=1^∞(n^2)/(n!)
(e) ∑n=1^∞(2^n n!)/(n^n)
(f) ∑n=1^∞(1)/((log n)^k) for some integer k.
(g) ∑n=1^∞(1)/(n(log n)^2)
(h) ∑n=1^∞(1)/((log n)^n)
(i) ∑n=1^∞((-1)^n)/((log n)^n)

Added by Pablo B.

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