Rewrite this as a sum?

Question

Rewrite sigma from n=2 to infinity of n(n-1)a_nx^n-2 as a sum whose generic term involves xⁿ.

Answer: sigma from n=0 to infinity of (n+2)(n+1)a_n+2xⁿ

Please help me step by step.

Answer 1

Hi Sun,  about this problem, assume that m ranges from 0 to ∞.

Let n=m+2.  This is correct since when m=0 we have n=2, when m=1, we have n=3, and so on.

Then, you can write your summation as

Σ_n=2^∞ (n)(n-1)a_n xⁿ  =  Σ_m=0^∞ (m+2)(m+2-1)a_m+2x^m+2-2 = Σ_m=0^∞ (m+2)(m+1)a_m+2x^m

which is exactly your answer. I wrote m, instead of n,  but this does not change anything as n or m are just names. It is actually more precise to change the name of the index from n to m as in this way you show to someone following your argument what you did. Hope this helps.

Best Regards,

Maurizio