Infinite Series and Convergence Tests

Lecture 36 18.01 Fall 2006 Lecture 36: Infinite Series and Convergence Tests Infinite Series Geometric Series A geometric series looks like 1 + a + a2 + a3 + ... = S There’s a trick to evaluate this: multiply both sides by a: a + a2 + a3 + ... = aS Subtracting, (1 + a + a2 + a3 + · · · ) − (a + a2 + a3 + · · · ) = S − aS In other words, 1 = S − aS =⇒ 1 = (1 − a)S =⇒ S = 1 1−a This only works when |a| < 1, i.e. −1 < a < 1. a = 1 can’t work: 1 + 1 + 1 + ... = ∞ a = −1 can’t work, either: 1 − 1 + 1 − 1 + ... = � 1 1 = 2 1 − (−1) Notation Here is some notation that’s useful for dealing with series or sums. An infinite sum is written: ∞ � ak = a0 + a1 + a2 + ... k=0 The finite sum Sn = n � ak = a0 + ... + an k=0 is called the “nth partial sum” of the infinite series. 1 Lecture 36 18.01 Fall 2006 Definition ∞ � ak = s k=0 means the same thing as lim Sn = s, where Sn = n→∞ n � ak k=0 We say the series converges to s, if the limit exists and is finite. The importance of convergence is illustrated here by the example of the geometric series. If a = 1, S = 1 + 1 + 1 + ... = ∞. But S − aS = 1 or ∞−∞=1 does not make sense and is not usable! Another type of series: ∞ � 1 p n n=1 We can use integrals to decide if this type of series converges. First, turn the sum into an integral: � ∞ ∞ � 1 dx ∼ p n xp 1 n=1 If that improper integral evaluates to a finite number, the series converges. Note: This approach only tells us whether or not a series converges. It does not tell us what number the series converges to. That is a much harder problem. For example, it takes a lot of work to determine ∞ � 1 π2 = n2 6 n=1 Mathematicians have only recently been able to determine that ∞ � 1 3 n n=1 converges to an irrational number! Harmonic Series � ∞ ∞ � 1 dx ∼ n x 1 n=1 We can evaluate the improper integral via Riemann sums. We’ll use the upper Riemann sum (see Figure 1) to get an upper bound on the value of the integral. 2 Lecture 36 18.01 Fall 2006 y=⁄x 1 ½ 1 ½ 1 2 3 ⅓ Figure 1: Upper Riemann Sum. N � 1 dx 1 1 ≤ 1 + + ... + = sN −1 ≤ sN x 2 N −1 We know that � As N → ∞, ln N → ∞, so sN N dx = ln N x 1 → ∞ as well. In other words, ∞ � 1 n n=1 diverges. Actually, sN approaches ∞ rather slowly. Let’s take the lower Riemann sum (see Figure 2). y=⁄x ¼ ½ 1 2 ⅓ 3 4 Figure 2: Lower Riemann Sum. sN = 1 + � N N � 1 dx 1 1 ≤1+ = 1 + ln N + ... + =1+ n x 2 N 1 n=2 Therefore, ln N < sN < 1 + ln N 3 Lecture 36 18.01 Fall 2006 Integral Comparison Consider a positive, decreasing function f (x) > 0. (For example, f (x) = 1 ) xp �∞ � � ∞ �� � � � f (n) − f (x)dx� < f (1) � � � 1 n=1 So, either both of the terms converge, or they both diverge. This is what we mean when we say � ∞ ∞ � 1 dx ∼ p n xp 1 n=1 Therefore, ∞ � 1 diverges for p ≤ 1 and converges for p > 1. p n n=1 Lots of fudge room: in comparison. ∞ � 1 √ 2 n + 10 n=1 diverges, because 1 1 1 √ ∼ 2 1/2 = 2 n (n ) n + 10 Limit comparison: � � If f (x) ∼ g(x) as x → ∞, then f (n) and g(n) either both converge or both diverge. What, exactly, does f (x) ∼ g(x) mean? It means that lim x→∞ f (x) =c g(x) where 0 < c < ∞. Let’s check: does the following series converge? ∞ � n √ 5 n − 10 n=1 n n 1 √ ∼ 5/2 = n−3/2 = 3/2 5 n n n − 10 Since 3 > 1, this series does converge. 2 4 Lecture 36 18.01 Fall 2006 Playing with blocks At this point in the lecture, the professor brings out several long, identical building blocks. Do you think it’s possible to stack the blocks like this? Top block is farther out than the bottom block. Figure 3: Collective center of mass of upper blocks is always over the base block. In order for this to work, you want the collective center of mass of the upper blocks always to be over the base block. The professor successfully builds the stack. Is it possible to extend this stack clear across the room? The best strategy is to build from the top block down. Let C0 be the left end of the first (top) block. Let C1 = the center of mass of the first block (top block). Put the second block as far to the right as possible, namely, so that it’s left end is at C1 (Figure 4). Let C2 = the center of mass of the top two blocks. Strategy: put the left end of the next block underneath the center of mass of all the previous ones combined. (See Figure 5). 5 Lecture 36 18.01 Fall 2006 2 1/2 1 C0 C1 C2 Figure 4: Stack of 2 Blocks. 2 1 2 3 1/2 1 1/3 C0 C1 C2 C3 Figure 5: Stack of 3 Blocks. Left end of block 3 is C2 = center of mass of blocks 1 and 2. C0 = 0 C1 = 1 C2 = 1 + Cn+1 = 1 2 nCn + 1(Cn + 1) (n + 1)Cn + 1 1 = = Cn + n+1 n+1 n+1 1 1 C3 = 1 + + 2 3 1 1 1 C4 = 1 + + + 2 3 4 1 1 1 1 C5 = 1 + + + + > 2 2 3 4 5 6 Lecture 36 18.01 Fall 2006 } n n+1 block center of mass of the first n blocks Figure 6: Stack of n + 1 Blocks. So yes, you can extend this stack as far (horizontally) as you want — provided that you have enough blocks. Another way of looking at this problem is to say N � 1 = SN n n=1 Recall the Riemann Sum estimation from the beginning of this lecture: ln N < SN < (ln N ) + 1 as N → ∞, SN → ∞. How high would this stack of blocks be if we extended it across the two lab tables here at the front of the lecture hall? The blocks are 30 cm by 3 cm (see Figure 7). One lab table is 6.5 blocks, or 13 units, long. Two tables are 26 units long. There will be 26 − 2 = 24 units of overhang in the stack. 3 cm 30 cm Figure 7: Side view of one block. If ln N = 24, then N = e24 . Height = 3 cm · e24 ≈ 8 × 108 m That height is roughly twice the distance to the moon. If you want the stack to span this room (∼ 30 ft.), it would have to be 1026 meters high. That’s about the diameter of the observable universe. 7