Question

4. a) Find the interval of convergence. 7m b) Determine whether the series in a) converges or diverges at the endpoints of the interval 5. Find the Maclauren series for f(x) = cos(2x). Include an expression for the neh term in the series.

Answer 1

a)

7t. 7t

Series Ratio Test

n1(n+1) an+1 an

n+1 = lim

n→00

1421 . 1

$=4\left|x\right|$

The sum converges for L < 1 therefore solve 4 < 1

$-\frac{1}{4}<x<\frac{1}{4}$

Hence, interval is

$\left (-\frac{1}{4},\frac{1}{4} \right )$

b)

Apply Alternating Series Test: converges 7t n= 7t

00 An 1 1 -| diverges n=1

$-\frac{1}{4}\le \:x<\frac{1}{4}$

Hence converges at endpoint x = -1/4 and diverges at endpoint x =1/4