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Worked solution

Explain when an arithmetic series can converge

Does an infinite arithmetic series have a sum to infinity? Why or why not?

CJC · Sequences and Series · Infinite arithmetic-series question

Answer

A nonzero arithmetic series does not have a finite sum to infinity. The all-zero case a=d=0a=d=0 converges to 00.

Compare your reasoning

Expand the AP partial sum

Sn=n2[2a+(n−1)d]=d2n2+(a−d2)n\begin{aligned}&S_n\\ & =\frac n2[2a+(n-1)d]\\ & =\frac d2n^2+\left(a-\frac d2\right)n\end{aligned}

The source starts from the AP formula and expands it as a polynomial in n.

Treat nonzero and zero differences separately

d≠0: ∣Sn∣→∞;d=0: Sn=nad\ne0:\ |S_n|\to\infty;\qquad d=0:\ S_n=na

For d>0 or d<0, the quadratic term dominates and the partial sums tend respectively to positive or negative infinity. When d=0, the quadratic term vanishes: S_n=na diverges for a≠0 but is always zero when a=0.

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Solution notes

The Teacher answer and summary table say an infinite arithmetic series has no sum to infinity. That statement needs the nonzero-series qualification: the all-zero AP is an exception. The source explanation in terms of a quadratic applies only when d≠0.