A Sequence of Zeros

★ 8/10calculusMonotonicitysequencesEstimationTelescoping

Problem

For n=1,2,…n = 1, 2, \dots, define

fn(x)=−1+x+x222+x332+⋯+xnn2.f_n(x) = -1 + x + \frac{x^2}{2^2} + \frac{x^3}{3^2} + \cdots + \frac{x^n}{n^2}.

1. Prove that for each n∈N∗n \in \mathbb{N}^* there exists a unique xn∈[23,1]x_n \in \left[\dfrac{2}{3}, 1\right] with fn(xn)=0f_n(x_n) = 0. 2. Prove that for every p∈N∗p \in \mathbb{N}^*, the sequence {xn}\{x_n\} from part 1 satisfies

0<xn−xn+p<1n.0 < x_n - x_{n+p} < \frac{1}{n}.

Answer

Solution

Difficulty8/10
Topicscalculus, Monotonicity, sequences, Estimation, Telescoping

Whiteboard

Your sketch is saved only in this browser. To share it, export your drawing as an image (whiteboard menu → Export as → PNG), then upload that image in the comments below.

Discussion

Ask questions, share alternate solutions, and use LaTeX freely.

0 comments
Log in to join the discussion.

No comments yet.