Question

Use the integral test to determine whether the following series converges or diverges.
∞
∑ 1÷(9+16k²)
k=1

Answer 1

Answer:

Divergence Test

For a series ∑n=1∞an to converge, the nth term an must satisfy an→0 as n→∞.

Therefore, from the algebraic limit properties of sequences,

limk→∞ak=limk→∞(Sk−Sk−1)=limk→∞Sk−limk→∞Sk−1=S−S=0.

Therefore, if ∑n=1∞an converges, the nth term an→0 as n→∞. An important consequence of this fact is the following statement:

Ifan↛0asn→∞,∑n=1∞andiverges.

5.8

This test is known as the divergence test because it provides a way of proving that a series diverges.

Step-by-step explanation:

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