Suppose is a sequence of positive numbers such that _ n d_ n+1 d_n =1

MATHEMATICS · Multi-part · AP · Calculus BC · Infinite Sequences and Series · Infinite Sequences and Series

Problem

Suppose

\{d_n\}

is a sequence of positive numbers such that

\lim_{n\to\infty}\frac{d_{n+1}}{d_n}=1.

Let

D=\sum_{n=1}^{\infty} d_n

. Respond to the items below. (a) Explain why knowing only that

\displaystyle \lim_{n\to\infty}\tfrac{d_{n+1}}{d_n}=1

is insufficient to determine whether

D

converges or diverges. Your explanation should reference at least one property of infinite series. (b) Provide an explicit example of a convergent series with positive terms whose successive-term ratio tends to

1

. State the general term of your series and verify both the ratio condition and convergence. (c) Provide an explicit example of a divergent series with positive terms whose successive-term ratio also tends to

1

. State the general term of your series and verify both the ratio condition and divergence. (d) Compare and contrast the two series presented in parts (b) and (c). Describe a key structural difference in their term formulas that accounts for the opposite convergence behavior despite having the same limiting ratio.

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