What is the historical motivation of infinite series? According to Wikipedia, they are arose separately by Newton, Leibniz and Somayaji.
I have not read Leibniz and Somayaji, but Newton himself says that his motivation is that one can solve "any equation" using infinite series, he means by this algebraic equations, differential equations (integrals is a special case of them) and many other types of functional equations. He demonstrates this with examples in his two letters to Leibniz, and then in expanded form in his papers on Analysis. What he uses is called Taylor series nowadays, and some more general series with non-integer powers of the variable.
He also explicitly mentions the analogy between the infinite decimal expansions and infinite series (decimal expansions are in fact a very special case).
It is clear that other mathematicians were also thinking along these lines. For example $\pi$ as a real number has a decimal expansion, in principle. But one cannot find all digits of this expansion explicitly. On the other hand various representations of $\pi$ by other explicit infinite expansions (series or products) were found. These expansions are more explicit than decimal representation because the pattern is clear in them.