Question

Determine whether the following series converges. Summation from k equals 2 to infinity (negative 1 )Superscript k Baseline StartFraction k squared minus 3 Over k squared plus 4 EndFraction Let a Subscript kgreater than or equals0 represent the magnitude of the terms of the given series. Identify and describe a Subscript k. Select the correct choice below and fill in any answer box in your choice. A. a Subscript kequals nothing is nondecreasing in magnitude for k greater than some index N. B. a Subscript kequals nothing and for any index​ N, there are some values of kgreater thanN for which a Subscript k plus 1greater than or equalsa Subscript k and some values of kgreater thanN for which a Subscript k plus 1less than or equalsa Subscript k. C. a Subscript kequals nothing is nonincreasing in magnitude for k greater than some index N. Evaluate ModifyingBelow lim With k right arrow infinitya Subscript k. ModifyingBelow lim With k right arrow infinitya Subscript kequals nothing Does the series​ converge?

Answer 1

`a_k=\left|(k^2-3)/(k^2+4)\right|;\text{ is nondecreasing for $k>2$}`

`\lim\limits_(k \to \infty) a_k =1`

The series does not converge

Explanation:

Given ...

`S=\displaystyle\sum\limits_(k=2)^(\infty){x_k}\\\\x_k=(-1)^k\cdot(k^2-3)/(k^2+4)\\\\a_k=|x_k|`

Find

whether S converges.

Solution

The (-1)^k factor has a magnitude of 1, so the magnitude of term k can be written as ...

`\boxed{a_k=1-(7)/(k^2+4)}`

This is non-decreasing for k>1 (all k-values of interest)

As k gets large, the fraction tends toward zero, so we have ...

`\boxed{\lim\limits_(k\to\infty){a_k}=1}`

Terms of the sum alternate sign, approaching a difference of 1. The series does not converge.