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An = (1+(-1)n)/(n2)

determine convergence or divergence of the sequence. If it
converges, find the limit.

User Benamir
by
7.9k points

1 Answer

6 votes

Final answer:

The sequence a_n = (1+(-1)^n)/(n^2) converges to 0 as the numerator alternates between 0 and 2 while the denominator, n squared, grows without bound, driving the fraction towards 0.

Step-by-step explanation:

We are tasked with determining the convergence or divergence of the sequence an = (1+(-1)n)/(n2). To do this, we can inspect the given formula for the sequence. Because the numerator alternates between 0 and 2 depending on whether n is odd or even, and the denominator grows without bound as n increases, it's clear that the overall fraction approaches 0 as n becomes large.

For even values of n, (-1)n = 1, thus the numerator becomes 1 + 1 = 2. For odd values of n, (-1)n = -1, thus the numerator becomes 1 - 1 = 0. Because n2 in the denominator gets larger and larger as n increases, regardless of the numerator being 2 or 0, the fraction's absolute value will get closer and closer to 0. This implies that the sequence converges to 0.

In conclusion, the limit of the sequence an as n tends to infinity is 0, indicating that the sequence indeed converges.

User Flapjack
by
7.5k points
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