If $\int_0^x\left(e^{-t^2}+\cos t\right)dt$ has the power series expansion $\sum_{n=1}^\infty a_n x^n$, then what is $a_5$ equal to?

Question

If $\int_0^x\left(e^{-t^2}+\cos t\right)dt$ has the power series expansion $\sum_{n=1}^\infty a_n x^n$, then what is $a_5$ equal to?

I know that the Cauchy integral test says that the power series converges if the integral converges and diverges if integral diverges. But this is the test for convergence and divergence, how can i find the fifth term of series?

Answer 1

Differentiating both sides, we obtain $e^{-x^2}+\cos x = \sum\limits_{n=1}^\infty n a_n x^{n-1}$, and we are interested in the term $5 a_5 x^4$.

Now, expand the left side using Taylor:

$$e^{-x^2}=1-x^2+\frac{x^4}{2}+O\left(x^6\right)$$

$$\cos x=1-\frac{x^2}{2}+\frac{x^4}{24}+O\left(x^6\right)$$

Finally, by the uniqueness of Taylor series, $\frac{1}{2}+\frac{1}{24}=5a_5$, which is easy to solve.