멱급수의 연산

[T] Evaluate the power series expansion $\ln (1+x)=\sum_{n=1}^{\infty }{(-1)}^{n-1}\frac{{x}^{n}}{n}$ at $x=1$ to show that $\ln (2)$ is the sum of the alternating harmonic series. Use the alternating series test to determine how many terms of the sum are needed to estimate $\ln (2)$ accurate to within 0.001, and find such an approximation.

Show that, up to powers x³ and y³, $E(x)=\sum_{n=0}^{\infty }\frac{{x}^{n}}{n!}$ satisfies $E(x+y)=E(x)E(y).$

Differentiate the series $E(x)=\sum_{n=0}^{\infty }\frac{{x}^{n}}{n!}$ term-by-term to show that $E(x)$ is equal to its derivative.

Show that if $f(x)=\sum_{n=0}^{\infty }{a}_{n}{x}^{n}$ is a sum of even powers, that is, ${a}_{n}=0$ if n is odd, then $F={\int }_{0}^{x}f(t)dt$ is a sum of odd powers, while if f is a sum of odd powers, then F is a sum of even powers.