Does this series $\cosh 1 + (\cosh 1-1)x^2+(\cosh1-1-\frac{1}{2!})x^4+(\cosh1-1-\frac{1}{2!}-\frac{1}{4!})x^6+\cdots$ converge?

Question

Can somebody help me with this?

$\cosh 1 + (\cosh 1-1)x^2+(\cosh1-1-\frac{1}{2!})x^4+(\cosh1-1-\frac{1}{2!}-\frac{1}{4!})x^6+\cdots$

Answer 1

For $|x|<1$ we can write

$$\begin{align} \sum_{n=1}^\infty\left(\cosh(1)- \sum_{k=1}^n \frac{1}{(2k-2)!}\right)x^{2n}&=\cosh(1)\frac{x^2}{1-x^2}-\sum_{k=1}^\infty \frac{1}{(2k-2)!}\sum_{n=k}^\infty x^{2n}\\\\ &=\cosh(1)\frac{x^2}{1-x^2}-\frac{1}{1-x^2}\sum_{k=1}^\infty \frac{x^{2k}}{(2k-2)!}\\\\ &=\cosh(1)\frac{x^2}{1-x^2}-\frac{x^2}{1-x^2}\sum_{k=1}^\infty \frac{x^{2k-2}}{(2k-2)!}\\\\ &=\frac{x^2(\cosh(1)-\cosh(x))}{1-x^2} \end{align}$$

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More generally, we have

$$\begin{align} \sum_{n=1}^N\left(\cosh(1)- \sum_{k=1}^n \frac{1}{(2k-2)!}\right)x^{2n}&=\cosh(1)\frac{x^2(1-x^{2N})}{1-x^2}-\sum_{k=1}^N \frac{1}{(2k-2)!}\sum_{n=k}^N x^{2n}\\\\ &=\cosh(1)\frac{x^2(1-x^{2N})}{1-x^2}-\frac{x^2}{1-x^2}\sum_{k=1}^N \frac{x^{2k-2}-x^{2N}}{(2k-2)!}\\\\ &=\frac{x^2}{1-x^2}\left(\cosh(1)-\sum_{k=1}^N\frac{x^{2k-2}}{(2k-2)!}\right)\\\\&+\frac{x^{2N+2}}{1-x^2}\left(\sum_{k=1}^\infty \frac{1}{(2k-2)!}-\cosh(1)\right)\\\\ &=\frac{x^2}{1-x^2}\left(\cosh(1)-\sum_{k=1}^N\frac{x^{2k-2}}{(2k-2)!}\right)\\\\&-\frac{x^{2N+2}}{1-x^2}\left(\sum_{k=N+1}^\infty \frac{1}{(2k-2)!}\right) \end{align}$$

Taking the limit as $N\to \infty$, we find that

$$\sum_{n=1}^\infty\left(\cosh(1)- \sum_{k=1}^n \frac{1}{(2k-2)!}\right)x^{2n}=\frac{x^2(\cosh(1)-\cosh(x))}{1-x^2}$$

for $|x|\ne 1$.