1. Prove or give counterexample. (a) If (rn) and (yn) diverge, then (rn + Yn) diverges. (b) If (1n) and (yn) diverge, then (r„Yn) diverges. (c) If (rn) and (xn + Yn) converge, then (yn) converges. (d) If (x„) and (rnYn) converge, then (y„) converges.
1. Prove or give counterexample. (a) If (rn) and (yn) diverge, then (rn + Yn) diverges. (b) If (1n) and (yn) diverge, then (r„Yn) diverges. (c) If (rn) and (xn + Yn) converge, then (yn) converges. (d) If (x„) and (rnYn) converge, then (y„) converges.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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I need help with #1. Thank you.
Transcribed Image Text:1. Prove or give counterexample.
(a) If (xn) and (Yn) diverge, then (xn + Yn) diverges.
(b) If (x„) and (Yn) diverge, then („Yn) diverges.
(c) If (xn) and (Tn + Yn) converge, then (yn) converges.
(d) If (x„) and („Yn) converge, then (yn) converges.
COs n
2. Use the definition of convergence to prove lim
= 0
n+0 n2 – n +1
3. Calculate the following limit analytically using theorems from the notes.
lim (Vn² + n – n)
4. Let xn → 0 and (yn) be bounded. Prove xnYn → 0.
5. Prove that r € E' iff there exists a sequence (xn) in E\{x} such that x, → r.
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