<s>[INST] Hint for Estimating an Infinite Series I am currently working through E. V. Shchepin's book Uppsala Lectures on Calculus: On Euler's Footsteps, and I've encountered a problem with estimating an infinite sum.
I need a hint estimating: S =  $\sum_{k=1}^\infty{1 \over2^kk}$. This problem appears early in the book in the section on unordered sums.
So far I have bounded S by comparing it to $\sum_{k=1}^\infty{1 \over2^k}$ = 1 
and $\sum_{k=1}^\infty{1 \over2^k 2^k}$ = ${1 \over 3}$
I've also tried various groupings of terms such as:
$\sum_{k=1}^\infty{1 \over2^kk}$ = ${1 \over 2}$ ${1 \over 1 }$ + ${1 \over 2^2}$${1 \over 2}$ + ${1 \over 2^3}$${1 \over 3}$ + ${1 \over 2^4}$${1 \over 4}$ +${1 \over 2^5}$${1 \over 5}$ +${1 \over 2^6}$${1 \over 6}$ + ${1 \over 2^7}$${1 \over 7}$ +${1 \over 2^8}$${1 \over 8}$ +$\cdots$
$\ge$ ${1 \over 2}$ ${1 \over 1 }$ + ${1 \over 2^2}$${1 \over 2}$ + (${1 \over 2^3}$${1 \over 4}$ + ${1 \over 2^4}$${1 \over 4}$) + (${1 \over 2^5}$${1 \over 8}$ +${1 \over 2^6}$${1 \over 8}$ + ${1 \over 2^7}$${1 \over 8}$ +${1 \over 2^8}$${1 \over 8}$) +$\cdots$
$=$ ${1 \over 2}$ + ${1 \over 2^3}$+ (${1 \over 2^5}$+ ${1 \over 2^6}$) + (${1 \over 2^8}$ +${1 \over 2^9}$ + ${1 \over 2^{10}}$ +${1 \over 2^{11}}$) +$\cdots$
which is $\sum_{k=1}^\infty{1 \over2^k}$ with some terms missing.  However, I can't find any pattern to the missing terms.
Any hints on how to proceed?
 [/INST] Write 
$$
\frac{1}{k} = 1 - \frac{1}{1\cdot 2} -\frac{1}{2\cdot 3} -\frac{1}{3\cdot 4} \cdots \frac{1}{(k-1)k}
$$
Then group terms to get
$$S = 1 \sum \frac{1}{2^k} - \frac{1}{1 \cdot 2} \sum \frac{1}{2^{k+1}} + \cdots 
$$
So for example your fifthapproximation will be
$$
1 - \frac{1}{2} \frac{1}{2} - \frac{1}{4} \frac{1}{6} - \frac{1}{8} \frac{1}{12} - \frac{1}{16} \frac{1}{20} = \frac{111}{160}
$$
which is accurate to three decimal digits.
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