If bj is positive for every j and if sum of bj from 1 to infinity converges then prove that the sum from 1 to infinity of 1/(1+bj) diverges
8)
If bj is positive for every j and if the sum of bj diverges then define sj= sum from l=1 to j of bl.
Discuss convergence or divergence for the series sum from j=1 to infinity of bj/sj .
24)
Let aj be a sequence of real numbers. Define
Prove that if lim aj = l as j tend to infinity then lim mj=l as j tends to infinity.
Give an example to show that the converse is not true.
28)
Let the sum of aj from j=1 to infinity be a divergent series of positive terms. Prove that there exist numbers bj, 0Similarly, let the sum of cj from j=1 to infinity be a convergent series of positive terms. Prove that there exist numbers dj, 0Part B
- Discuss the convergence or divergence for each of the following series: (for all of them the sum goes from j=1 to infinity)
5)
If bj is positive for every j and if the sum of bj from j=1 to infinity converges then prove that the sum from j=1 to infinity of bj/(1+bj) converges.
6) Let p be a polynomial with no constant term. If bj is positive for every j and if the sum of bj from j=1 to infinity converges then prove that the series the sum of p(bj) from j=1 to infinity converges.
13)
Examine the series 1/3+ 1/5 + 1/3² + 1/5² + 1/3^3 + 1/5^3 + 1/3^4 + 1/5^4 +…
Prove that the root test shows that the series converges while the ratio test give no information.
14)
Check that both the root test and the ratio test give no information for the series the sums j=1 to infinity of 1/j , the sum of 1/j² from j=1 to infinity. However, one of these series is divergent and the other is convergent.
18) Prove theorem 4.4
Consider the series Saj from j=1 to infinity of nonzero terms. If lim sup of absolute value of aj to the 1/j power is strictly greater than 1, then the series diverges.
19) Prove theorem 4.5
Consider the series Saj from j=1 to infinity. If there is an N>0 such that absolute value of aj+1/aj is greater than or equal to 1, for all j greater than or equal to N then the series diverges.
21)
Let the sum of aj from 1 to infinity and the sum of bj from 1 to infinity be convergent series of positive real number. Discuss convergence of the sum of aj*bj from j=1 to infinity
22) What can you say about the convergence or divergence of the of j=1 to infinity of ?
23)
If bj>0 and the sum of bj from j=1 to infinity converges then prove that the sum from j=1 to infinity of ((bj)^0.5 )*1/(j^a) converges for any a>1/2 . Give an example to show that the assertion is false if a=1/2.
25) Imitate the proof of the root test to give a direct proof of the ratio test.
29)
Let be series with positive summands. Prove that if there is a constant C>0 such that 1/C £aj/bj £C for all j large then either both series diverge or both series converge
Part C
2)
Let p be a polynomial with integer coefficients. Let b1³b2³…³0 and assume that bj approaches 0. Prove that if (-1)^p(j) is not always positive and not always negative then in fact it will alternate in sign so that will converge.
3)
If bj >0 for every j and if converges then prove that ² converges. Prove that the assertion is false if the positivity hypothesis is omitted. How about third powers?
7)
Assume that the sum of j=1 to infinity of bj is an absolutely convergent series of real nulbers. Let sj = .(I meant b subscript l). Discuss convergence or divergence for the series sum of j=1 to infinity of sj*bj . Discuss convergence or divergence for the series
10) Let the sum of bj from j=1 to infinity be a conditionally convergent series of real numbers. Let b be a real number. Prove that there is a rearrangement of the series that converges to b. (Hint: First observe that the positive terms of the given series must form a divergent series. Also, the negative terms form a divergent series. Now build the rearrangement by choosing finitely many positive terms whose sum “just exceeds” b. Then add on enough negative terms so that the sum is “just less than” b. Repear this oscillatory procedure.)
12) Follow these steps to give another proof of the Alternating Series test: a) Prove that the odd partial sums form an increasing sequence; b) Prove that the even partial sums form a decreasing sequence; c) Prove that every even partial sum majorizes all subsequent odd partial sums; d)Use a pinching principle.
26)
Prove that and are both divergent series.
II) Chapter 3 exercises
4) Let {aj} a sequence of real or complex nulbers with limit a and {bj} be a sequence of real or complex numbers with limit b. Then
Prove the following statement
- The sequence {aj + bj} converges to a+b
- If bj¹0 for all j and b¹0 then the sequence aj/bj converges to a/b.
5) Prove the following result, which we have used without comment in the text: let S be a set of real numbers which is bounded above and let t= sup S. For every e>0 there is an element s ÎS such that t- e
6) Provide the details of the following remark: The Bolzano-Weierstrass theorem is a generalization of our result from the last section about monotone increasing sequences which are bounded above (resp. monotone decreasing sequences which are bounded below). For such a sequence is surely bounded above and below (why?). So it has a convergent subsequence. And thus it follows easily that the entire sequence converges.
14) Prove that (1 + x/j)^j converges to for any real number x.
17) Discuss the convergence of the sequence {(j^j)/(2j)!} j going from 1 to infinity.
22) Let {aj} be a sequence of complex numbers. Suppose that for every pair of integers N> M>0 it holds that Prove that {aj} converges.
23) Let a1, a2 >0 and for j ³ 3 define aj = aj-1 + aj-2 . Show that this sequence cqnnot converge to a finite limit.
24) Suppose a sequence {aj} has the property that for every natural number N there is a j sub N such thqt aj sub N = aj sub N+1 =…= aJ sub N+ N. In other words, the sequence has arbitrarily long repetitive strings. Does it follow that the sequence converges?
27)
Prove the pinching principle:
Let {aj} {bj} and {cj} be sequences of real numbers satisfying aj £ bj £ cj for every j.
If lim a j = lim cj = a as j tends to infinity for some real number a then limb j = a as j tends to infinity.
29)
Consider the sequence given by aj = + Use a picture to give a convincing argument that the sequence {aj} converges. The limit number is called ? . This number was first studied by Euler. It arises in many different context in analysis and number theory. Show that for some universal constant C>0.
Part B
7)
Let {aj} be a sequence of real or complex numbers. Suppose that every subsequence has itself a subsequence which converges to a given number a. Prove that the full sequence converges to a.
8) Supply the details this:
The sequence {j²} converges to +¥ . The sequence {-2j+18} converges to -¥. The sequence {j+ j*(-1)^j} has no infinite limit and no finite limit. However, the subsequence {0,0,0,…} converges to 0 and the subsequence {4,8,12,…} converges to +¥.
25)
Give an example of a single sequence of rational numbers with the property that for every real number a there is a subsequence converging to a.
26)
Let S ={0,1,1/2,1/3,1/4,…}. Give an example of a sequence {aj} with the property that for each s Î S there is a subsequence converging to s, but no subsequence converges to any limit not in S.
28)
Give another proof of the Bolzano-Weierstrass theorem as follows. If {aj} is a bounded sequence let bj = inf{aj, aj+1,…} . Then each bj is finite, b1 £ b2£…,and {bj} is bounded above. Use this proposition : If {aj} is a monotone increasing sequence which is bounded above aj £ M for all j then {aj} is convergent. If {bj} is a monotone decreasing sequence which is bounded below bj ³ K> - ¥ for all j then {bj} is convergent.
Part C:
11) Provide the details of the assertion , made in the text, that the sequence {cos j} is dense in the interval [-1,1].
15) Discuss the convergence of the sequence {(1/j)^1/j } j going from 1 to infinity
16) Find lim sup and lim inf of the sequences {} and {