Question

Given that the series kcoskt kº +2 k=1 converges, suppose that the 3rd partial sum of the series is used to estimate the sum of the series. Which of the following values gives the best bound on the remainder (error) for this approximation?

a. 1/2
b. -3 / 29
c. 2/ 33
d. 1/5

Answer 1

c

Explanation:

Given that:

`\sum \limits ^(\infty)_(k=1) (kcos (k\pi))/(k^3+2)`

since cos (kπ) =
`-1^k`

Then, the series can be expressed as:

`\sum \limits ^(\infty)_(k=1) ((-1)^kk))/(k^3+2)`

In the sum of an alternating series, the best bound on the remainder for the approximation is related to its
`(n+1)^{th` term.

∴

`\sum \limits ^(\infty)_(k=1) ((-1)^((3+1))(3+1)))/((3+1)^3+2)`

`\sum \limits ^(\infty)_(k=1) ((-1)^((4))(4)))/((4)^3+2)`

`= (4)/(64+2)`

`=(2)/(33)`