Question

Find an expression for the general term of each of the series below. cn​=0 for all n - The sum starts with n=0

x³ex²=∑n=0[infinity]​cn​(x)=x³+x⁵+x⁷/2!+x⁹/3!+....​

Answer 1

The general term of the given series is a power series where the odd powers of x are added with coefficients determined by factorial expressions.

Step-by-step explanation:

The given series is a power series where the general term is defined by cn = 0 for all n. The sum in the series is given by x³ex² = ∑n=0[infinity]​cn​(x) = x³ + x⁵ + x⁷/2! + x⁹/3! + ....

Since cn = 0 for all n, the series only contains odd powers of x. The power of x in each term is increasing by 2 starting from 3. The coefficients of the terms are determined by the factorial of the exponent divided by the corresponding factorial index.

Therefore, the general term of the series is given by cn = {x³, x⁵, x⁷/2!, x⁹/3!, ...}.