Let's have some fun:

Express$$\sum_{n=2}^{\infty}n(n-1)c_nx^{n-2}+\sum_{n=0}^{\infty}c_nx^{n+1}$$as a single power series.

Question

Let's have some fun:

Express$$\sum_{n=2}^{\infty}n(n-1)c_nx^{n-2}+\sum_{n=0}^{\infty}c_nx^{n+1}$$as a single power series.

across · Answer

Hint: Make the powers of x in each series be "in phase."

across · Answer

Hint: Make both summation indices start with the same number.

across · Answer

$$2c_2+\sum_{n=3}^{\infty}n(n-1)c_nx^{n-2}+\sum_{n=0}^{\infty}c_nx^{n+1}$$

across · Answer

$$k_1=n-2,$$$$k_2=n+1.$$$$2c_2+\sum_{k_1=1}^{\infty}(k_1+2)(k_1+1)c_{k_1+2}x^{k_1}+\sum_{k_2=1}^{\infty}c_{k_2-1}x^{k_2}.$$

across · Answer

$$2c_2+\sum_{k=1}^{\infty}\left[(k+1)(k+2)c_{k+2}+c_{k-1}\right]x^k.$$

anonymous · Answer

$$
\begin{align}
&\sum_{2 \le n \le \infty}n(n-1)c_nx^{n-2}+\sum_{0 \le n \le \infty}c_nx^{n+1}\
&=\sum_{2 \le n+2 \le \infty}(n+2)((n+2)-1)c_{n+2}x^{(n+2)-2}+\sum_{0 \le n \le \infty}c_nx^{n+1}\
&=\sum_{0 \le n \le \infty}(n+2)(n+1)c_{n+2}x^{n}+\sum_{0 \le n \le \infty} c_nx^{n+1}\
&=(0+2)(0+1)c_{0+2}x^{0}+\sum_{1 \le n \le \infty}(n+2)(n+1)c_{n+2}x^{n}+\sum_{0 \le n \le \infty}c_nx^{n+1}\
&=2c_2+\sum_{1 \le n+1 \le \infty}((n+1)+2)((n+1)+1)c_{(n+1)+2}x^{n+1}+\sum_{0 \le n \le \infty}c_nx^{n+1}\
&=2c_2+\sum_{0 \le n \le \infty}(n+3)(n+2)c_{n+3}x^{n+1}+\sum_{0 \le n \le \infty}c_{n}x^{n+1}\
&=2c_2+\sum_{0 \le n \le \infty}(c_n+(n+3)(n+2)c_{n+3})x^{n+1}
\end{align}
$$

Typically, power series begin at $0$.