Solutions - 0180-02

(a) The series has positive terms.

$$\frac{1}{n^{1/3}+2^n}<\frac{1}{2^n}$$

But ${\displaystyle \sum \frac{1}{2^n}}$ converges because it is geometric with $r=1/2$. By the DCT, the original series must converge.

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(b) The series has positive terms.

$$\frac{\sqrt{k}}{k-1}>\frac{\sqrt{k}}{k}=\frac{1}{k^{1/2}}$$

But ${\displaystyle \sum \frac{1}{k^{1/2}}}$ diverges because it is a $p$-series with $p=1/2<1$. By the DCT, the original series must diverge.