Summing Maclaurin Series for x^2

AI Thread Summary
To find the sum of the series ∑(x^(2k)/k!), the user initially attempted to transform known Taylor series but encountered issues with the factorials and the signs in the series. The key challenge was that the series involved k! in the denominator rather than (2k)!. However, the user successfully resolved the problem by substituting x with x^2 in the Taylor series expansion of e^x, leading to the desired sum. This approach effectively aligns the series with the correct factorial structure and maintains the positive terms throughout. The solution demonstrates the utility of manipulating existing series to derive new results.
JakeD
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Homework Statement


How do I find the sum of \sum\frac{x^{2k}}{k!}?

The Attempt at a Solution


I tried transforming various known Taylor series, such as sin x, e^x, and so on, but they didn't fit for 2 reasons:
1. In all of them, the degree of the factor equals the power of x. i.e. if you have x^2k in the nominator, then you have (2k)! in the denominator, whereas here, you have x^2k in the nominator, while having k! (not (2k!)) in the denominator.

2. In sin x, you have alternating pluses and minuses, while in the required sum, they are all pluses.Any help will be appreciated
 
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OK, found the solution.

By replacing x with x^2 in the Taylor series of e^x, I get the desired sum.
 
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