Question

Determine whether the series is convergent or divergent by expressing Sn as a telescopic sum (as in example 7). If it is convergent, calculate its sum

Answer 1

Convergent

cos 1 − 1

Explanation:

∑ (cos 1/n² − cos 1/(n+1)²)

= lim(n→∞) [(cos 1 − cos 1/4) + (cos 1/4 − cos 1/9) + ... + (cos 1/n² − cos 1/(n+1)²)]

= lim(n→∞) [cos 1 − cos 1/(n+1)²]

= cos 1 − cos 0

= cos 1 − 1

The series converges to cos 1 − 1.