Shifting the Index of a Series of Real Numbers

In the setting of The Real Numbers: Standing Notation and Background, let $(a_{k})_{k\in\mathbb{N}}$ be a sequence of real numbers and let $(b_{k})_{k\in\mathbb{N}}$ be the sequence given by

$$b_{k}=a_{k+1}\qquad(k\in\mathbb{N}).$$

Convergence of a series of real numbers and its sum are as defined there.

(i) The series $\sum_{k=1}^{\infty}a_{k}$ converges if and only if the series $\sum_{k=1}^{\infty}b_{k}$ converges, and in that case

$$\sum_{k=1}^{\infty}a_{k}=a_{1}+\sum_{k=1}^{\infty}b_{k}.$$