What do I do to implement the #x^2# into this series? #x^2sum_(n=0)^oo(na_nx^(n-1))#

1 Answer
Oct 27, 2017

# sum_(n=0)^oo (na_nx^(n+1))#

Explanation:

Let:
# S = x^2sum_(n=0)^oo(na_nx^(n-1)) #

If unclear as to the effect then the best option to expand a few terms of the summation:

# S = x^2{0a_0x^(-1) + 1a_1x^0 + 2a_2x^1 + 3a_3x^2 + 4a_4x^3 + ...}#
# \ \ = {0a_0x^(1) + 1a_1x^2 + 2a_2x^3 + 3a_3x^4 + 4a_4x^5 + ...}#

Then we can put it the series back into "sigma" notation:

# S = sum_(n=0)^oo (na_nx^(n+1))#