Problem: Find the sum of the series

Solution: This series contains parts of the form c^k, but there are more of them, so first we should tidy up the given expression.

This is not a geometric series because of that k! in the denominator, so there is no way to sum it up using the geometric approach, and there also seems no way to change it into a difference in order to apply the telescopic series trick. So there are two possibilities. Either we will be able to use the power series approach, or to show that the given series diverges. However, note that if we ignore the k! for a second, then we get a geometric series with q = −2/3, thus |q| < 1 and therefore the geometric series is convergent. Terms of our series are even smaller, which suggests that it also converges and therefore we do have to sum it up. We need to find a well-fitting Taylor expansion. Do we know some power series that has k! in the denominator? We do, the series for the exponential.

There is a little problem, though, that the series for exponential starts with k = 0, while our series starts at 1. This is simple to fix, we fill in the missing part. Since the given series can be made to fit to this power series, we can use this expansion to sum up the given series.

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