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Proposition 7.2.5. Let man be a formal series of real numbers. Then Enm an converges if and only if, for every real number & > 0, there exists an integer N≥m such that q 2₁s |n=p Proof. See Exercise 7.2.2. an ≤e for all p, q≥ N. TA This Proposition, by itself, is not very handy, because it is not so easy to compute the partial sums E-pan in practice. However, it has a number of useful corollaries. For instance: Corollary 7.2.6 (Zero test). Let En-m an be a convergent series of real numbers. Then we must have limnoo an = 0. To put this another way, if limnoo an is non-zero or divergent, then the series En man is divergent.

Proposition 7.2.5. Let man be a formal series of real numbers. Then Enm an converges if and only if, for every real number & > 0, there exists an integer N≥m such that q 2₁s |n=p Proof. See Exercise 7.2.2. an ≤e for all p, q≥ N. TA This Proposition, by itself, is not very handy, because it is not so easy to compute the partial sums E-pan in practice. However, it has a number of useful corollaries. For instance: Corollary 7.2.6 (Zero test). Let En-m an be a convergent series of real numbers. Then we must have limnoo an = 0. To put this another way, if limnoo an is non-zero or divergent, then the series En man is divergent.

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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Please use Proposition 7.2.5 to prove Corollary 7.2.6

 

 

Proposition 7.2.5. Let man be a formal series of real numbers. Then Enm an converges if and only if, for every real number ɛ > 0, there exists an integer N≥m such that q |n=p Proof. See Exercise 7.2.2. an & for all p, q≥ N. 01.S.T. This Proposition, by itself, is not very handy, because it is not so easy to compute the partial sums E-p an in practice. However, it has a number of useful corollaries. For instance: Corollary 7.2.6 (Zero test). Let Enm an be a convergent series of real numbers. Then we must have limno An 0. To put this another way, if limn→∞ an is non-zero or divergent, then the series En man is divergent. BELO Proof. See Exercise 7.2.3. = IS.7 col imun leve
Transcribed Image Text:Proposition 7.2.5. Let man be a formal series of real numbers. Then Enm an converges if and only if, for every real number ɛ > 0, there exists an integer N≥m such that q |n=p Proof. See Exercise 7.2.2. an & for all p, q≥ N. 01.S.T. This Proposition, by itself, is not very handy, because it is not so easy to compute the partial sums E-p an in practice. However, it has a number of useful corollaries. For instance: Corollary 7.2.6 (Zero test). Let Enm an be a convergent series of real numbers. Then we must have limno An 0. To put this another way, if limn→∞ an is non-zero or divergent, then the series En man is divergent. BELO Proof. See Exercise 7.2.3. = IS.7 col imun leve
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