Kummer's Series Transformation

Let $\sum_{k=0}^\infty a_k=a$ and $\sum_{k=0}^\infty c_k=c$ be convergent series such that

$$\lim_{k\to\infty} {a_k\over c_k} = \lambda\not =0.$$

Then

$$a =\lambda c+\sum_{k=0}^\infty \left({1-\lambda {c_k\over a_k}}\right)a_k.$$

References

Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 16, 1972.