7: You should get

So it looks like the sine expansion after all. Two things need to be done, fixing the exponent in the power of x and then changing indexing so that the factorial is applied to 2k + 1. What shift is necessary? You get a good hint if you first try to start with 2k − 1 and rewrite it in such a way that +1 appears. Since shift in indexing is going to change the exponent, it is obvious in which order one should do these things: First shift the index, then fix the exponent. Do not forget to make sure that after shifting the index, the indexing starts at the right place for the sine expansion.

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