lim Bounded above by M Converges to L M

Transcription

1 Calculus 2 MONOTONE SEQUENCES If for all we say is nondecreasing. If for all we say is increasing. Similarly for nonincreasing decreasing. A sequence is said to be monotonic if it is either nondecreasing or nonincreasing strictly monotonic if in addition consecutive terms are never equal. If the inequalities hold for all we say the sequence is eventually monotonic. Recall that dropping a finite number of terms from a sequence does not affect its convergence property (whether it converges or diverges). The convergence property of is the same as the convergence property of its tail. MONOTONE CONVERGENCE THEOREM (MCT) If a sequence is monotone nondecreasing it satisfies one of the convergence properties below. 1. It increases without bound is therefore divergent. The lim 2. It is bounded above by ( for all ) therefore lim. The sequence converges to a number less than or equal to. M a n Increases without bound Diverges to infinity lim n Bounded above by M Converges to L M n MONOTONE CONVERGENCE THEOREM (MCT) If a sequence is monotone nonincreasing it satisfies one of the convergence properties below. 1. It decreases without bound is therefore divergent. The lim. 2. It is bounded below by ( for all ) therefore lim. The sequence converges to a number greater than or equal to. Eventual monotonicity is all that we really need for MCT because the convergence property of is the same as the convergence property of its tail. A direct consequence of MCT is that any decreasing sequence of positive terms is convergent (or any nonincreasing sequence of nonnegative terms) because positive terms are bounded below by zero. IF A SEQUENCE IS POSITIVE AND CONVERGES TO ZERO THAT DOES NOT MEAN IT IS DECREASING Example: Constant sign convergence to zero do not imply monotonicity. sin / is positive lim 0 but the terms are not eventually decreasing they are oscillating. IF A SEQUENCE IS POSITIVE AND DIVERGES TO INFINITY THAT DOES NOT MEAN IT IS INCREASING Example: Constant sign divergence to infinity do not imply monotonicity. when is odd when is even. is positive lim because both the even numbered terms the odd numbered terms diverge to infinty, but the terms are not eventually increasing. They alternately increase decrease. If is even then 1. If is odd then 1. HdMonotoneSeq.docx Prof. L.A. Month Page 1 of 8

2 y lim S 0 lim x We prove a sequence is increasing by proving that difference 0. ratio 1 (only for sequence of positive terms) derivative Let, 0 (if the derivative exists) Similarly for decreasing. Example: 1 Is the sequence increasing or decreasing eventually? 1. The sequence is increasing. 0. The sequence is increasing., 0. The sequence is increasing. It is easy to see in this case lim The terms are increasing to 1 from below. Be careful. GIVEN A SEQUENCE OF POSITIVE TERMS WHERE AND ARE POSITIVE. DOES NOT IMPLY THE SEQUENCE IS DECREASING. IT MERELY IMPLIES THE TERMS ARE LESS THAN 1 Example: The sequence is decreasing. 2 1 The sequence is decreasing bounded below by zero. converges by MCT. In this case it is easy to see lim Example: 2 HdMonotoneSeq.docx Prof. L.A. Month Page 2 of 8

3 2 The sequence 2 0 if 2. The sequence is decreasing eventually. is decreasing bounded below by zero. 2 converges by MCT. In this case it is easy to see lim 2 0. Example: 1. The sequence is increasing. Also 1. The sequence is bounded above by 1. The monotone convergence theorem tells us the sequence converges to a limit 1 because the sequence is increasing bounded above by 1. In this case we can easily find the limit of the sequence. lim 2 31 Example: The sequence is decreasing bounded below by zero. The sequence converges by MCT. In this case we can easily find the limit of the sequence. 2 lim 31 Let's look at the list of the previous examples with labels for increasing for decreasing. is, is, is, is, is We already noted that given a sequence of positive terms where are positive. does not imply the sequence is decreasing. It merely implies the terms are less than 1. You might think that if we also have then the sequence is decreasing. But looking at the above examples you see that this is not the case. In fact you might find it surprising that have different monotone properties. Are there sufficient condtions for monotonicity? Consider a sequence where are positive differentiable functions defined for all the values of agree with the values of, for. The sequence is decreasing if only if ln ln HdMonotoneSeq.docx Prof. L.A. Month Page 3 of 8

4 Theorem: Given a sequence where are positive differentiable functions for all agrees with for. The sequence is decreasing if only if (increasing iff ln ln ) Re-examining the two examples we see by factoring that although both sequences converge to 2/3 the first is always less than 2/3 the second is always greater than 2/3. So maybe the results are not so surprising after all!! 2/3 / / 2/3. / Example:! We can t let!!!! because! doesn t exist!!! 1 for 1. The sequence is nonincreasing for 1 (The 1 2 terms are equal.) The sequence 2 is nonincreasing. The sequence 2 is also bounded below!! by zero because all the terms are positive. By the monotone convergence theorem 2 converges (to a limit 0).! Example:!!!!! 1 for 9. The sequence is nonincreasing for 9. (The 9 th 10 th terms are equal.) The sequence 10 is nonincreasing. The sequence 10 is also bounded below!! by zero because all the terms are positive. By the monotone convergence theorem 10 converges (to a limit 0).! Proof: lim 0! Let lim lim The monotone convergence theorem tells us exists. from our previous calculation 10 lim 1 lim 10 1 = lim 1 lim 10 lim lim 0 means! grows faster than 10 (or any base to the for that matter).! Example: lim 0 (for any real fixed ) Proof:! 1. If 0 then lim 0.! 2. lim 0, 0.! Proof: Let! HdMonotoneSeq.docx Prof. L.A. Month Page 4 of 8

5 !!! 1 for 1.! The sequence is decreasing bounded below by zero.! By the monotone convergence theorem lim 0. Let lim lim The monotone convergence theorem tells us exists. 1 from our previous calculation lim 1 lim lim lim lim 0! Proof:!!! lim lim 0! lim 0 Squeeze theorem! lim 0 means! grows faster than! for any real fixed x. FACTORIAL TRUMPS EXPONENTIAL Example: What about lim It is easier to consider!!! lim 0! lim lim? 0. all the terms are positive, therefore lim! We can also show the sequence! is monotone decreasing.!! The sequence!!!!! 1. 1! is decreasing bounded below by zero. By MCT converges (to a limit 0).! Example: Another way to prove lim 0 lim lim 0. The monotone convergence theorem tells us exists.. from our previous calculation Take lim of both sides. which has solution 0. HdMonotoneSeq.docx Prof. L.A. Month Page 5 of 8

6 (We have used the result that 1 1 to evaluate lim 1 lim! 0 means that grows faster than!. lim 1 1 ) Be sure to see the hout on factorials bounds. For n large enough ln! 1 fixed The point is that can be arbitrarily large can be arbitrarily small positive. For example ln. eventually For n large enough ln! The inequality follows by noting that therefore The bounds ln! 1 apply to the growth rates as well. This mean means in addition to ln. eventually it is also true that lim 0. which is equivalent to the much stronger result that eventually ln. for any arbitrarily small positive Similarly in addition to! 10 eventually it is also true that! lim which is equivalent to the much stronger result that eventually! 10 for any arbitrarily large positive We close with two well known results! ~ 2 Stirling s approximation for large which means lim! 1! for 1 Proof: for 1 ln! ln2 3 4 ln2ln3ln4ln We can bound ln! by by bounding the area under the graph of ln above below by the sum of the areas of rectangles of height ln width 1. The area of each rectangle is ln the sum of the areas of the rectangles is ln 2 ln 3 ln 4 ln ln!. See the figures.! HdMonotoneSeq.docx Prof. L.A. Month Page 6 of 8

7 ln x ln x ln n ln n ln 2 ln 3 ln n-1 ln n-1 n x ln 2 ln 3 ln n-1 x n n+1 ln ln! ln ln 1ln! 1 ln 1 ln 1ln!ln1!! 1!! Example: lim! /! /!! / / / / lim lim / lim ( is continuous at 1) lim 1 / 1 1! lim Squeeze theorem. (Use L Hôpital s rule to evaluate lim 1 / ln1 1/ ln1 lim ln11/ ln1 lim 0 HdMonotoneSeq.docx Prof. L.A. Month Page 7 of 8

8 lim 1 / lim / / 1 is continuous at 0 ) Example: The sequence! is decreasing!!!! / 1. (The last inequality follows from the fact that lim is increasing. The proof that 1 1 is increasing is a famous proof by Riemann using the binomial theorem.) It follows from MCT that the sequence! converges. Finding the limit of the sequence follows from Stirling's formula. FYI lim lim!! 1! lim lim! lim lim / 1 lim / 1 0 lim 0 1 lim 1 lim lim 1 lim is fixed! 0 any! lim 0! lim 0 lim any! lim lim! lim 2! lim 1! /! / For homework problems see HdMonotoneSeq.docx Prof. L.A. Month Page 8 of 8