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Worked solution

Resolve the constant cases left out of the table

What happens to the infinite series when (a) d=0d=0 in an AP, and (b) r=1r=1 in a GP?

CJC · Sequences and Series · AP/GP constant-case callout

(a) Zero AP difference

Answer

For a>0a>0 the partial sums tend to positive infinity; for a<0a<0 they tend to negative infinity. If a=0a=0, the all-zero series converges to zero.

Sum the constant terms

d=0⟹un=a,Sn=nad=0\quad\Longrightarrow\quad u_n=a,\qquad S_n=na

This callout is printed in both copies without a worked answer; the cases are authored working.

How did this part go?

(b) Unit GP ratio

Answer

A nonzero GP with r=1r=1 diverges, with the sign of infinity determined by aa. The all-zero extension converges to zero.

Use the constant sum instead of dividing by zero

r=1⟹un=a,Sn=nar=1\quad\Longrightarrow\quad u_n=a,\qquad S_n=na

The ordinary finite-GP quotient cannot be substituted at r=1. The source convention excludes zero terms from a GP, so a=0 is an extension rather than a nonzero GP.

How did this part go?

Common mistakes

Convergence requires a finite limit. A limit of positive or negative infinity is divergence. A constant term sequence can converge while its series diverges.