Chapter 11 - Series Solutions to Linear Differential Equations - 11.1 A Review of Power Series - Problems - Page 730: 13

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Shift $n \rightarrow m+2$: $\sum n(n-1)a_{n-1}x^{n-2}=\sum (m+1)(m+2)a_{m+1}x^m$ Shift $n \rightarrow m+1$: $\sum na_{n}x^{n-1}=\sum (m+1)a_{m+1}x^m$ Hence, $\sum n(n-1)a_{n-1}x^{n-2}+\sum na_{n}x^{n-1}=\sum (m+1)(m+2)a_{m+1}x^m +\sum (m+1)a_{m+1}x^m\\ =\sum (m+1)(m+1+2)a_{m+1}x^m\ =\sum (m+1)(m+3)a_{m+1}x^m$