Serie alternada. d) \( \sum_{n=1}^{\infty}(-1)^{n+1} \frac{n}{n^{2}+1} \) \( \sum_{n=1}^{n} \frac{1}{n} n^{n / 2} \)

77-78 Use l'Hospital's Rule to evaluate the limit. 77. \( \lim _{x \rightarrow 0} \frac{1}{x^{2}} \int_{0}^{x} \frac{2 t}{\sqrt{t^{3}+1}} d t \)

\( \int _{}^{}\frac{1}{e^{x^{2}}}e^{\int _{}^{}2 d x} d x \)

\( \int _{}^{}\frac{1}{e^{x^{2}}}e^{\int _{}^{}2 d x} d x \)

Calculate the Riemann sum for the function \( f(x) = x^2 \) on the interval [1, 3] using 4 subintervals with left endpoints.

Use polynomial long division to evaluate the integral of \( \frac{x^4 + 6x^3 + 4x^2 + 8}{x^2 + 2} \) for integration.