In this post we shall discuss an ODE with initial conditions to derive a power series expansion for near . Let us consider the following initial value problem

.

It is quite clear that the function , satisfies the ODE above and since the coefficients are variable it is natural to look for a power series solution . Differentiation term by term yields

and .

Since , , our initial value problem will be equivalent to

.

Clearly, , , and for , . We easily obtain that for ,

and .

Therefore, this implies

.

The above formula serves as a good ingredient in evaluating series involving the central binomial coefficient like

This series converges at by Raabe’s test, and for it is uniformly convergent by the Weierstrass M-test.

Now, substitute , , and we get

.

Integrating from to , we have

.

On the other hand, Wallis’ formula (integral form) tells us that , and thus we finally obtain Euler’s celebrated

.

**Remark**. The ideas presented above have been generalized in this short paper.

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