Using power series representation, calculate ∑n=1∞n2n3n

eliaskidszs

eliaskidszs

Answered

2022-01-06

Using power series representation, calculate
n=1n2n3n

Answer & Explanation

Jim Hunt

Jim Hunt

Expert

2022-01-07Added 45 answers

Recall that, in general,
1+x+x2+=11x, |x|<1
Moreover, power series can be differentiated term by term. So, differentiating both sides of the equation above we get
1+2x+3x2+=1(1x)2, |x|<1
Now, multiplying both sides by x leads to
x+2x2+3x3+=n=1nxn=x(1x)2, |x|<1
However, in this case x=23<1, so simply substitute x=23 in formula above.
Marcus Herman

Marcus Herman

Expert

2022-01-08Added 41 answers

Basically you start with
x=0xn=11x
And then you do all the mathematical operations such as ddx on both sides until you get the form you want. For example, the first derivative will give you
x=1nxn1=1(1x)2
A popular second step you can do from there is multiply both sides by x, which gives you
x=1nxn=x(1x)2
karton

karton

Expert

2022-01-11Added 439 answers

First observe that your series is the special case of
n=1nzn
with z=23, which has radius of convergence R=1
By using the Cauchy product on n=0zn=11z we get
(11z)2=(n=0zn)2=n=0(k=0n)zn=n=0(n+1)zn=n=1nzn1
and after multiplying by z
n=1nzn=z(1z)2
For z=23 we get 23(123)2=23=6

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