For what values of $x$ does the following series converge?

Question

I need to find the values of $x$ for which the MacLarurin expansion of $4\cos(x)\ln(1+x)$ converges. The series that I obtained up to the $8^{th}$ order is: $$T_p=4x-2x^2-\frac23x^3+\frac3{10}x^5-\frac14x^6+\frac{31}{140}x^7-\frac{37}{180}x^8 + O(x^9)$$ However, I cannot seem to find a pattern in order to express the series as an infinite sum, and then use the ratio test (or other appropiate tests) to find for which values of $x$ it converges. Any ideas?

Answer 1

Hint

Taylor expansion of $\cos(x)$ is valid for any value of $x$ while Taylor expansion of $\log(1+x)$ is valid for $|x|<1$.

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