Question

Write out the first five terms of the sequence [(−1)n−1 /(n+10)²][infinity]n=1[(−1)n−1(n+10)²]n=1[infinity], determine whether the sequence converges, and if so find its limit.

Enter the following information for an=(−1)n−1(n+10)²an=(−1)n−1(n+10)².
a1=
a2=
a3=
a4=
a5=
Does the sequence converge = _______ ( yes/no)

Answer 1

The first five terms of the sequence are 1/121, -1/144, 1/169, -1/196, 1/225. The terms of the sequence approach 0 as n increases, and thus the sequence converges to a limit of 0.

Step-by-step explanation:

The given sequence is an = (−¹)n−1 / (n+10)². To find the first five terms of this sequence, we need to plug in the values of n from 1 to 5:

• a1 = (−¹)0 / (1+10)² = 1 / 121
• a2 = (−¹)1 / (2+10)² = − 1 / 144
• a3 = (−¹)2 / (3+10)² = 1 / 169
• a4 = (−¹)3 / (4+10)² = − 1 / 196
• a5 = (−¹)4 / (5+10)² = 1 / 225

To determine whether the sequence converges, we can observe that the denominators are squares of increasing positive integers and hence the terms are getting closer to 0 as n increases. Since the terms are decreasing in magnitude and alternating in sign, the sequence converges to a limit of 0.

Does the sequence converge = yes