Answer
This equation is valid.
Work Step by Step
1. Substitute each 'k' from the second expression on the left for a 'k - 1':
$$\sum^{\infty}_{k-1=0} a_{k-1}x^{k-1+1} = \sum^{\infty}_{k=1}a_{k-1}x^k$$
2. Separate the first expression on the left from its first term:
$$\sum^{\infty}_{k=0}a_{k+1}x^k = a_1x^0 + \sum^{\infty}_{k=1}a_{k+1}x^k $$
3. Sum these two series:
$$a_1 + \sum^{\infty}_{k=1}a_{k+1}x^k + \sum^{\infty}_{k=1}a_{k-1}x^k = a_1 + \sum^{\infty}_{k=1}(a_{k+1}x^k + a_{k-1}x^k)$$ $$a_1 + \sum^{\infty}_{k=1}(a_{k+1} + a_{k-1})x^k$$
This is exactly the expression on the left, therefore, they are equivalent.
