Final answer:
The question involves finding the sum of a complex infinite series. Without a recognizable pattern or more context, we cannot directly apply the binomial theorem or other series expansions to find the sum. Further information is required to proceed with the problem.
Step-by-step explanation:
The question asks to find the sum of the infinite series 1 + 4x + 8x² + 13x³ + 19x⁴ + ... , assuming it converges for |x| < 1. This series does not immediately appear to fit the pattern of a geometric or arithmetic series or a known series expansion. However, we are given a hint related to the binomial theorem and powers of numbers.
To find the sum of the series, we need to look for a pattern that could relate it to a known series, possibly manipulating the terms or comparing it to the expansion of a known function or applying a transformation similar to the binomial theorem that involves expanding a function in terms of powers of x.
Given that no straightforward pattern is apparent, we cannot directly apply a simple formula such as those derived from the binomial theorem or other series expansions without further information. Therefore, at this stage, we cannot find the sum of this series without additional details or context that might indicate a specific method to transform the series into a sumable form.