📚 How to test for convergence in alternating and recursively-defined sequences

What you'll need:

Alternating sequence: If lim_(n→∞)⁡〖|a_n |=0〗, then lim_(n→∞)⁡〖a_n=0〗

Recursively sequence: If {a_n } is convergent, then lim_(n→∞)⁡〖a_(n+1)=lim_(n→∞)⁡〖a_n 〗 〗

• If the sequence converges, then all terms in the sequence approach the same value, which we call L.

Q1. Does the following sequence converge or diverge?

a_n={(−1)^n⋅1/n^2 }

Q2. Suppose the sequence defined by a_(n+1)=1/( 5 ) (a_n+6) with a_(n=1)=2 converges. Find the limit.

Q3. Suppose the sequence defined by a_(n+1)=1/( 2 ) (a_n+4) with a_1=1 converges. Find the limit.