Problem 74 Hard Difficulty

Let $ I_n = \displaystyle \int_0^{\frac{\pi}{2}} \sin^n x dx $.
(a) Show that $ I_{2n + 2} \le I_{2n + 1} \le I_{2n} $.
(b) Use Exercise 50 to show that
$$ \frac{I_{2n + 2}}{I_{2n}} = \frac{2n + 1}{2n + 2} $$
(c) Use parts (a) and (b) to show that
$$ \frac{2n + 1}{2n + 2} \le \frac{I_{2n + 1}}{I_{2n}} \le 1 $$
and deduce that $ \lim_{n\to\infty}\frac{I_{2n + 1}}{I_{2n}} = 1 $.
(d) Use part (c) and Exercises 49 and 50 to show that
$$ \lim_{n\to\infty} \frac{2}{1} \cdot \frac{2}{3} \cdot \frac{4}{3} \cdot \frac{4}{5} \cdot \frac{6}{5} \cdot \frac{6}{7} \cdots \cdots \frac{2n}{2n - 1} \cdot \frac{2n}{2n + 1} = \frac{\pi}{2} $$
This formula is usually written as an infinite product:
$$ \frac{\pi}{2} = \frac{2}{1} \cdot \frac{2}{3} \cdot \frac{4}{3} \cdot \frac{4}{5} \cdot \frac{6}{5} \cdot \frac{6}{7} \cdots $$
and is called the Wallis product.
(e) We construct rectangles as follows. Start with a square of area 1 and attach rectangles of area 1 alternately beside or on top of the previous rectangle (see the figure). Find the limit of the ratios of width to height of these rectangles.

Answer

a) $\int_{0}^{\pi / 2} \sin ^{2 n+2} x d x \leq \int_{0}^{\pi / 2} \sin ^{2 n+1} x d x \leq \int_{0}^{\pi / 2} \sin ^{2 n} x d x$
So $, I_{2 n+2} \leq I_{2 n+1} \leq I_{2 n}$
b) Figure out the expression for $I_{2 n+2} .$ The part of $I_{2 n+2}$ that is the same as
$I_{2 n}$ cancels out easily.
c) 1
d) $\lim _{n \rightarrow \infty} \frac{1}{(2 n+1)}\left(\frac{2^{n}(n !)}{(2 n) !}\right)^{2}\left(2^{n}(n !)\right)^{2}=1$
$\lim _{n \rightarrow \infty} \frac{1}{(2 n+1)}\left(\frac{2^{n}(n !)}{(2 n) !}\right)^{2}\left(2^{n}(n !)\right)^{2}=\frac{\pi}{2}$
e) $\frac{\pi}{2}$

Related Courses

Calculus 2 / BC

Calculus: Early Transcendentals

Chapter 7

Techniques of Integration

Section 1

Integration by Parts

Discussion

You must be signed in to discuss.

Top Calculus 2 / BC Educators

Watch More Solved Questions in Chapter 7

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

Problem 6

Problem 7

Problem 8

Problem 9

Problem 10

Problem 11

Problem 12

Problem 13

Problem 14

Problem 15

Problem 16

Problem 17

Problem 18

Problem 19

Problem 20

Problem 21

Problem 22

Problem 23

Problem 24

Problem 25

Problem 26

Problem 27

Problem 28

Problem 29

Problem 30

Problem 31

Problem 32

Problem 33

Problem 34

Problem 35

Problem 36

Problem 37

Problem 38

Problem 39

Problem 40

Problem 41

Problem 42

Problem 43

Problem 44

Problem 45

Problem 46

Problem 47

Problem 48

Problem 49

Problem 50

Problem 51

Problem 52

Problem 53

Problem 54

Problem 55

Problem 56

Problem 57

Problem 58

Problem 59

Problem 60

Problem 61

Problem 62

Problem 63

Problem 64

Problem 65

Problem 66

Problem 67

Problem 68

Problem 69

Problem 70

Problem 71

Problem 72

Problem 73

Problem 74

Video Transcript

The problem is that I am asleep. Go to integral from zero to pi were to sign ax to empower the X A is showed that I, too am pass to its Morris and I to impress one. It's more is that it? I'm too late. The reason use or a reason is fi ox. It's small land created on zero. So we have signed two on two Actually small sign who in hostel on X Smaller than sign Teo. Thanks. So we have I tuned us to its more than hi to muscle on this more than I to im is use exercise fifteen to show that I too am plus two over high too. And his people to to cross one over two m pass too. In exercise fifty, we have an integral from zero to pine over too. Sign likes to power the ox is you caught minus one over a pen into girl from zero to a pirate to sign ice two a month Last power jacks So replace and buy to him last too. And I have this result apart. See yes part a party to show that I too have a plus one over to you and it's great. Tremulous one over two. M plus two and smaller than one. With this part, this is a direct result from a on DH for this part you can say hi to on us one over. I too. And is Caritas on I chew on past too. Over high Toolan. Aye. Aye. Then this is just to, um Paswan over to n plus two. And by squeeze Syria. This Let me help. This one is the Coto one. What Bang goes to infinity. And this is also one so limiter off. This one is what What the you need to prove the limit of this number is they go to pirate too. Here for simplicity with Whitey thiss number as a Am I right, sis? Or I can write. This's a two and no. Then what? And just infinity. We need to prove limiter a to an auto pilot too. By the result of exercise forty nine, we have I to m plus one is culture two times weren't home six times the times Who am I over three times off? I'm seven times that last one. Hi. Five exercise And what nine? We have to end. Yes, they couldn't. You Juan comes three times. I'm starting to him months, Juan over two times four times three times to him. Hands high over too. So we can find eye to us one over I am. It's just a two and times to our high. Then things as a result say limit and gostin affinity. I too am plus blanc over. I am by two and two and this is the comte one. So we have the layman tough a two and is the auto. Hira too was the last part for me. Let's look at some rectangles were constructed rectangles as follows starting with squired all I rear one and attach rectangles of area one ultimately beside on top of the previous rectangles. Example This is one Want one on DH? This third graf is want this one? Another pump is this one Long wants is this is my half. Just, uh, force oneness. Here we go. The tangos here. This Juan Juan one half here. This is to over three on the fifth one. No, like this. Is this the one half one one one to over three. This is and our three Is it is thiss sinus that this is three over, so we can. Two question is find its limit of the ratios of wise to a height of dis rectangles. So for one, the first one the first one, the re shows why wise or hide its frequent. Who won second? Juan Wai's over Hyde. It's difficult to to over one third one this wise our hut. This because too two over one class. One half This is culture to our You really want to do this for a green, we can lie. This's to over one times two over three and the fourth one wass over hard. It's the culture ain't over green over. You really want to. This is sixteen over nine, so we can write that this is too over one times, two over three times for a war. Three. I'm surface yes, los over heart. This is and overthe three over one plus a half Los three over eight, and you can tack. This is equal to to O. R. One times to over three Ham's flora work three times or over just the comm pyre. This result with heart. See, you can see this is exactly remember off a two, and it's only now that limit is because to pine over, too, by the result of part of

Wen Z.

Related Courses

Calculus 2 / BC

Calculus: Early Transcendentals

Chapter 7

Techniques of Integration

Section 1

Integration by Parts

Problem 74 Hard Difficulty

Answer

Related Courses

Calculus: Early Transcendentals

Related Topics

Discussion

Top Calculus 2 / BC Educators

Catherine R.

Anna Marie V.

Kayleah T.

Joseph L.

Calculus 2 / BC Courses

Integration Review - Intro

Definite vs Indefinite

Recommended Videos

Watch More Solved Questions in Chapter 7

Video Transcript

Calculus: Early Transcendentals

Catherine R.

Anna Marie V.

Kayleah T.

Joseph L.

Integration Review - Intro

Definite vs Indefinite