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Question

# Use the test of your choice to determine whether the following series converge.$\sum _ { k = 1 } ^ { \infty } \left( \frac { 1 } { k } + 2 ^ { - k } \right)$

Solution

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Step 1
1 of 2

Let $a_k=\dfrac {1} {k}, \ b_k=2^{-k}.$

Then,

$\sum a_k$

diverges (Harmonic series) and

$\sum b_k$

converges (Geometric series, $r=1/2$).

By Theorem 10.8, Property (4)

$\quad \Rightarrow \quad \sum (a_k+b_k) \quad \text { diverges }$

So,

$\sum_{k=1}^{\infty} \left( \frac {1} {k} +2^{-k}\right) \quad \text { diverges}$

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