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How to sum up this series :

$2C_o + \frac{2^2}{2}C_1 + \frac{2^3}{3}C_2 + \cdots + \frac{2^{n+1}}{n+1}C_n$

Any hint that will lead me to the correct solution will be highly appreciated.

EDIT: Here $C_i = ^nC_i $

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    This looks like it will be a dupe. I am unable to find it though. Anyone?2010-12-11

4 Answers 4

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REMARK $\ $ The various approaches are all equivalent. Namely, suppose that we desire to prove without calculus the identity arising from integrating the binomial formula, viz. $\rm (1 + x)^{n+1}\ =\ 1 + \sum_{k=1}^{n+1}\: \frac{n+1}{k+1} {n\choose k}\ x^{k+1}$

Comparing coefficients reduces it to the identity

$\rm \quad\quad\ {n+1 \choose k+1}\ =\ \frac{n+1}{k+1} {n\choose k} $

which is precisely the identity employed in Moron's "calculus free" approach.

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    @Moro$n$: Perhaps you read too much into the *name* "calculus $f$ree" - which I used merely to be consistent with terminology in other posts here. My point was to illustrate what the integration amounts to $f$rom a generating function perspective. Your proof can be derived completely *mechanically* from this equivalence.2010-12-11
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For an elementary method which does not use calculus:

Notice that $\displaystyle \dfrac{{n \choose k}}{k+1} = \dfrac{1}{n+1} {n+1 \choose k+1}$

Thus your sum is

$\sum_{k=0}^{n} \dfrac{1}{n+1} {n+1 \choose k+1} 2^{k+1} = \dfrac{\sum_{k=0}^{n+1} {n+1 \choose k} 2^k -1}{n+1} = \dfrac{3^{n+1} - 1}{n+1}$

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    Oh, yes! Now I even remember where I saw this first: In "Problem solving through problems", by Loren C. Larson.2010-12-11
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Hint: Use binomial expansion for $(1+x)^n$ and integrate once. Then choose an appropriate value for $x$.

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    Opps earlier it was a typo.Btw your hints always just made my day:)2010-12-11
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Let's assume $C_i=\binom ni$. I'll give a solution that is not precalculus level. Consider first the equality $ (1+x)^n=C_0+xC_1+x^2C_2+\dots+x^nC_n. $ This is the binomial theorem.

Integrate from 0 to t. On the left hand side we get $\frac{(1+t)^{n+1}-1}{n+1}$ and on the right hand side $\sum \frac1{i+1}t^{i+1}C_i$.

Now set $t=2$, and a bit of algebra gives you the answer you want.

Pretty sure there is an elementary approach as well.

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    What I said is that I haven't started off with calculus ;-)2010-12-11