Question

Use Direct Comparison Test to determine whether the series is convergent or divergent:

Answer 1

A series that has a finite sum is called convergent series. Otherwise, are divergent.

• Direct Comparison Test

Supposing that aₙ ≥ 0 and bₙ ≥ 0 for all values of n.

• Converges:, If aₙ ≤ bₙ for all values of n and

$\Sigma_{n=0}^{\infty}b_n$

converges, then the series:

$\Sigma^{\infty}_{n=0}a_n$

also converges.

• Diverges: ,If aₙ ≥ bₙ for all values of n and

$\Sigma^{\infty}_{n=0}b_n$

diverges, then the series:

$\Sigma^{\infty}_{n=0}a_{n}$

also diverges.

Our series:

$\Sigma_{n=0}^{\infty}\frac{6+\sin(n)}{n^2}$

Applying the comparison we can see that the series converges.

Answer: converge.