Consider the series and define for

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

Problem

Consider the series U=∑n=2∞sin⁡ ⁣(1n)n\displaystyle U=\sum_{n=2}^{\infty} \dfrac{\sin\!\bigl(\tfrac{1}{n}\bigr)}{\sqrt{n}} and define q(x)=sin⁡(1/x)xq(x)=\dfrac{\sin(1/x)}{\sqrt{x}} for x≥2x\ge 2. Answer the following questions. (a) Prove that q(x)>0q(x)>0 for every x≥2x\ge 2. Include any necessary discussion of its limiting behavior as x→∞x\to\infty. (b) Show that there exists an integer M≥2M\ge 2 such that qq is decreasing on [M,∞)[M,\infty). Provide either a derivative-based argument or a comparison argument, and identify an explicit value of MM that works. (c) Evaluate the improper integral ∫M∞sin⁡(1/x)x dx\displaystyle \int_{M}^{\infty} \dfrac{\sin(1/x)}{\sqrt{x}}\,dx or bound it sharply enough to determine whether it is finite. Clearly justify every step in your estimation or calculation. (d) Decide whether the series UU converges or diverges, citing your results from parts (a)–(c). (e) A student proposes using the ratio test on the series UU but is unable to reach a conclusion. Explain in detail why the ratio test is inconclusive in this context and why the approach you used in part (d) is more suitable.

Related theory

Browse the problem library