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Theorem 5. Abel's Lemma, or Partial Summation Given Sn = a1 + a2 + + an ... (a) a1v1+...+ a„Vn = $1(v1 – v2) + ...+Sn–1(Vn–1– Vn)+SnVn (b) If m < a1 + .. + an < M Vn, and vn is positive and decreasing, then mvị < a¡v1 + ... + anVn < Mv1. + ɑnVn < Mv1.

Theorem 5. Abel's Lemma, or Partial Summation Given Sn = a1 + a2 + + an ... (a) a1v1+...+ a„Vn = $1(v1 – v2) + ...+Sn–1(Vn–1– Vn)+SnVn (b) If m < a1 + .. + an < M Vn, and vn is positive and decreasing, then mvị < a¡v1 + ... + anVn < Mv1. + ɑnVn < Mv1.

Advanced Engineering Mathematics
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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Prove (b)

Theorem 5. Abel's Lemma, or Partial Summation Given Sn ai + a2 + + an ... (a) a1v1+...+anVn = 81(v1 – V2)+...+8n-1(Vn-1– Vn)+SnVn (b) If m < ai+ decreasing, then mvi < a¡v1 + (c) If in (b) |sn|< M Vn, then |a1v1 + ... + an Vn| < Mv1 Vn. | ... + an < M Vn, and v, is positive and ... + a„Vn < Mv1.
Transcribed Image Text:Theorem 5. Abel's Lemma, or Partial Summation Given Sn ai + a2 + + an ... (a) a1v1+...+anVn = 81(v1 – V2)+...+8n-1(Vn-1– Vn)+SnVn (b) If m < ai+ decreasing, then mvi < a¡v1 + (c) If in (b) |sn|< M Vn, then |a1v1 + ... + an Vn| < Mv1 Vn. | ... + an < M Vn, and v, is positive and ... + a„Vn < Mv1.
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