Consider this following proof of a combinatorial identity i.e $n2^{n-1}=\sum_{k=1}^{n}k \binom{n}{k}$
Proof:
$(1+x)^n=\sum_{k=1}^{n} \binom{n}{k} x^k$
Differentiating both sides wrt x gives
$n(1+x)^{n-1}=\sum_{k=1}^{n}k \binom{n}{k} x^{k-1}$
$x:=1 \Rightarrow n2^{n-1}=\sum_{k=1}^{n}k \binom{n}{k}$
How can this be justified for example if i say $x^2 -1=0$ Differentiating both sides wrt x gives $2x=0 \Rightarrow x=0$ which is not true :/.
Sorry if this sound foolish am i missing something?
A question about derivatives
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Hmm..Hammer...Treat everything as nail
Re: A question about derivatives
In the first case both side is function, and true for all real x; where it's not true for $x^2-1=0$Asif Hossain wrote: ↑Mon May 10, 2021 10:20 pmConsider this following proof of a combinatorial identity i.e $n2^{n-1}=\sum_{k=1}^{n}k \binom{n}{k}$
Proof:
$(1+x)^n=\sum_{k=1}^{n} \binom{n}{k} x^k$
Differentiating both sides wrt x gives
$n(1+x)^{n-1}=\sum_{k=1}^{n}k \binom{n}{k} x^{k-1}$
$x:=1 \Rightarrow n2^{n-1}=\sum_{k=1}^{n}k \binom{n}{k}$
How can this be justified for example if i say $x^2 -1=0$ Differentiating both sides wrt x gives $2x=0 \Rightarrow x=0$ which is not true :/.
Sorry if this sound foolish am i missing something?
-
- Posts:194
- Joined:Sat Jan 02, 2021 9:28 pm
Re: A question about derivatives
O thanks for pointing that out~Aurn0b~ wrote: ↑Mon May 10, 2021 11:26 pmIn the first case both side is function, and true for all real x; where it's not true for $x^2-1=0$Asif Hossain wrote: ↑Mon May 10, 2021 10:20 pmConsider this following proof of a combinatorial identity i.e $n2^{n-1}=\sum_{k=1}^{n}k \binom{n}{k}$
Proof:
$(1+x)^n=\sum_{k=1}^{n} \binom{n}{k} x^k$
Differentiating both sides wrt x gives
$n(1+x)^{n-1}=\sum_{k=1}^{n}k \binom{n}{k} x^{k-1}$
$x:=1 \Rightarrow n2^{n-1}=\sum_{k=1}^{n}k \binom{n}{k}$
How can this be justified for example if i say $x^2 -1=0$ Differentiating both sides wrt x gives $2x=0 \Rightarrow x=0$ which is not true :/.
Sorry if this sound foolish am i missing something?
Hmm..Hammer...Treat everything as nail