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August 29th, 2017, 10:34 PM   #1
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Exclamation Show that the sum is equal to the value



I'm not quite sure where to go with this one. The problem hints at using the binomial expansion, but I'm struggling to see how it will help me here. I tried writing out a couple of terms to no avail as well as expanded the combinatoric portion. I would greatly appreciate any strategies you may have to offer.
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August 29th, 2017, 11:05 PM   #2
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Write the binomial expansion of $(1 + x)^n$, then list the differences between that polynomial and the left-hand side of the equation you are given to prove.
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August 29th, 2017, 11:48 PM   #3
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Quote:
Originally Posted by John Travolski View Post


I'm not quite sure where to go with this one. The problem hints at using the binomial expansion, but I'm struggling to see how it will help me here. I tried writing out a couple of terms to no avail as well as expanded the combinatoric portion. I would greatly appreciate any strategies you may have to offer.
$(1+x)^n = \displaystyle \sum \limits_{k=0}^{n} \dbinom{n}{k} x^k$

differentiate w/respect to $x$

$n(1+x)^{n-1} = \displaystyle \sum \limits_{k=1}^{n}k \dbinom{n}{k} x^{k-1}$

let $x=1$

$n2^{n-1} = \displaystyle \sum \limits_{k=1}^{n} k \dbinom{n}{k}$
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