prove that 4 Prove the Monotone Convergence theorem, i.e. (a) if a monotone sequence (an) is bounded then (an) is convergent, and (b) if a monotone sequence (an) is NOT bounded then (an) is divergent. 5 Let can be a series with positive terms such n=1 Син that the sequence (anti) of successive terms is Convergent, with limit L. (a) Prove that if OELI then Σan is divergent. 6 Using the Comparison Test, the Ratio Test and the Alternating Series Test as needed, determine if each of the following series. Σan of real numbers is absolutely convergent, conditionally Convergent or (a) n² Ž n = 1 हात divergent. (b) (-1)+1 n=2 n n² = 1 (c) Σ (-2)² n = on! ①(a) Let rεR, o≤r<1. Prove directly from defintions. 2 = r ++²+3+... that the series r = 1 is convergent with sum 1-2 +p+.... (b) Prove that the harmonic series 2 ½ is divergent. นะเ n ② (a) Define what is meant by the following: (1) the sequence (an) is convergent. (ii) the sequence (an) is Cauchy. (ii) the sequence (an) is bounded. (b) Prove directly from definitions that a Cauchy Sequence is bounded. ③ (a) Prove diredly from definitions that if a Cauchy Sequence (an) has a convergent subsequence then (an) is convergent. (6) Use the results of 2(b) and 3(a) to deduce that a Counchy sequence is convergent. You may assume the Bolzano- Weierstrass Theorem.
prove that 4 Prove the Monotone Convergence theorem, i.e. (a) if a monotone sequence (an) is bounded then (an) is convergent, and (b) if a monotone sequence (an) is NOT bounded then (an) is divergent. 5 Let can be a series with positive terms such n=1 Син that the sequence (anti) of successive terms is Convergent, with limit L. (a) Prove that if OELI then Σan is divergent. 6 Using the Comparison Test, the Ratio Test and the Alternating Series Test as needed, determine if each of the following series. Σan of real numbers is absolutely convergent, conditionally Convergent or (a) n² Ž n = 1 हात divergent. (b) (-1)+1 n=2 n n² = 1 (c) Σ (-2)² n = on! ①(a) Let rεR, o≤r<1. Prove directly from defintions. 2 = r ++²+3+... that the series r = 1 is convergent with sum 1-2 +p+.... (b) Prove that the harmonic series 2 ½ is divergent. นะเ n ② (a) Define what is meant by the following: (1) the sequence (an) is convergent. (ii) the sequence (an) is Cauchy. (ii) the sequence (an) is bounded. (b) Prove directly from definitions that a Cauchy Sequence is bounded. ③ (a) Prove diredly from definitions that if a Cauchy Sequence (an) has a convergent subsequence then (an) is convergent. (6) Use the results of 2(b) and 3(a) to deduce that a Counchy sequence is convergent. You may assume the Bolzano- Weierstrass Theorem.
Chapter9: Sequences, Probability And Counting Theory
Section9.4: Series And Their Notations
Problem 10TI: Determine whether the sum of the infinite series is defined. 24+(12)+6+(3)+
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Transcribed Image Text:prove
that
4 Prove the Monotone Convergence theorem, i.e.
(a) if a monotone sequence (an) is bounded then (an)
is convergent, and
(b) if a monotone sequence (an) is NOT bounded then
(an) is divergent.
5 Let can be a series with positive terms such
n=1
Син
that the sequence (anti) of successive terms is
Convergent, with limit L.
(a) Prove that if OEL<I then Σan is conungent.
(b) Prove that if L>I then Σan is divergent.
6 Using the Comparison Test, the Ratio Test and the
Alternating Series Test
as needed, determine if
each of the following series.
Σan of real
numbers is absolutely convergent, conditionally
Convergent or
(a) n²
Ž
n = 1
हात
divergent.
(b) (-1)+1
n=2
n
n² = 1
(c) Σ (-2)²
n = on!
Transcribed Image Text:①(a) Let rεR, o≤r<1. Prove directly from defintions.
2 = r ++²+3+...
that the series
r = 1
is convergent with sum
1-2
+p+....
(b) Prove that the harmonic series 2 ½ is divergent.
นะเ
n
② (a) Define what is meant by the following:
(1) the sequence (an) is convergent.
(ii) the sequence (an) is Cauchy.
(ii) the sequence (an) is bounded.
(b) Prove directly from definitions that a Cauchy
Sequence is bounded.
③ (a) Prove diredly from definitions that if a Cauchy
Sequence (an) has a convergent subsequence
then (an) is convergent.
(6) Use the results of 2(b) and 3(a) to deduce
that a
Counchy sequence is convergent. You
may assume the Bolzano- Weierstrass Theorem.
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