Consider each infinite series given below

MATHEMATICS · Subjective · AP · Calculus BC · Infinite Sequences and Series · Infinite Sequences and Series

Problem

Consider each infinite series given below. For every series, analyze its behavior at infinity. (a) ∑k=1∞5(13)k−1\displaystyle \sum_{k=1}^{\infty} 5\left(\tfrac13\right)^{k-1} (b) ∑k=2∞8(32)k−2\displaystyle \sum_{k=2}^{\infty} 8\left(\tfrac32\right)^{k-2} (c) ∑k=0∞(−4)( ⁣−12)k\displaystyle \sum_{k=0}^{\infty} (-4)\left(\!-\tfrac12\right)^{k} (d) ∑k=1∞1k\displaystyle \sum_{k=1}^{\infty} \dfrac1k For each series, determine whether it converges or diverges and provide a complete explanation that references the defining characteristics of the terms involved. Conclude by comparing the overall long-term behavior of the four series and summarizing the key differences you observe.

Related theory

Browse the problem library