Taylor and Maclaurin series flashcards

Question 1

What is a power series centered at $a$?

Accepted Answer

An infinite series of the form $$\sum_{n=0}^{\infty} c_n (x-a)^n,$$ where $c_n$ are constants and $a$ is the center.

Question 2

State the Maclaurin series for $\cos x$.

Accepted Answer

$$\cos x=\sum_{n=0}^{\infty}(-1)^n\frac{x^{2n}}{(2n)!}=1-\frac{x^2}{2!}+\frac{x^4}{4!}-\cdots$$ (converges for all $x$).

Question 3

Use the ratio test to find the radius of convergence of $\sum_{n=1}^{\infty} \frac{(x-2)^n}{n}$.

Accepted Answer

Let $a_n=\frac{(x-2)^n}{n}$. Then $$\left|\frac{a_{n+1}}{a_n}ight|=\left|\frac{(x-2)^{n+1}}{n+1}\cdot\frac{n}{(x-2)^n}ight|=|x-2|\cdot\frac{n}{n+1}	o |x-2|.$$ Converges if $|x-2|<1$, so $R=1$.

Taylor and Maclaurin series flashcards

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