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Numerade Educator



Problem 50 Hard Difficulty

Around 1910, the Indian mathematician Srinivasa Ramanujan discovered the formula

$ \frac {1}{\pi} = \frac {2 \sqrt{2}}{9801} \displaystyle \sum_{n = 0}^{\infty} \frac {(4n)!(1103 + 26390n)}{(n!)^4396^{4n}} $

William Gosper used this series in 1985 to compute the first
17 million digits of $ \pi . $
(a) Verify that the series is convergent.
(b) How many correct decimal places of $ \pi $ do you get if you use just the first term of the series? What if you use two terms?


a. convergent.
b 6 decimal places; 15 decimal places


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Video Transcript

you have this Siri's representation for one over pi and for part A. We'd like to show that the Siri's converges, and so let's go ahead and resolve this for part A. To do this to show a convergence. Let's use the ratio test. Where will go ahead and set this term right here? Evil tee am. So let me not right that whole thing out So and will be this large fraction. But we will still have to do a bit of writing here. So notice that if we look at the absolute value of a N plus one over N, we could drop the absolute value because the ends are always positive. So in this case, let's go ahead and read. Deal with this numerator here. So we replace all the ends with the N plus one. So the end factorial to the fourth becomes n plus one factorial to the fourth. And then we have three ninety six and then four and then and plus one noticed the parentheses here and then we'LL go ahead and divide that by a n So when we divide by an, we could go ahead and just flip this term here and multiplied to the red So that'LL be as usual is you've noticed when you use the ratio test in the past Here, there we go. So we flipped and over and then multiplied it so eventually will want to take a limit of this. But until we get to that point way, want to simplify as much as we can? So first thing I'LL do here is if we will get let's maybe going a different color here blue. So this term right here, this would be This is also for en plus four factorial and that would be the product of all the numbers all the way out to Foreign Plus four. But before you get there youcause foreign, foreign plus one and so on until you reach foreign Plus Four. And the reason I'm writing these additional terms out is because this first terms here all the way up to four n That's foreign factorial. So we could go ahead and cancel this with this and we're left over with this in the numerator. Similarly, I can go ahead and make it distribute this twenty six thousand term through the parentheses. So let me go ahead and do that You'LL see why shortly This is a good idea so that one of these terms should have it end So this one will have an end And what else can we do here? So now we look at again Going to the blue Let's look it and plus one factorial So the fourth so n plus one factorial using the same ideas before this can be written his n factorial times and plus one However, we also haven't in factorial over here and we're raising to the fourth power. So we raises to the fourth we get in factorial to the fourth and then and plus once of the fourth. And so we could cancel this in factorial to the forth with that one and that'LL leave us with the n plus one So the fourth in the denominator Now what else do we have? So if we distribute this foreign plus one through this is foreign plus four So we could cancel the foreign term here with the one down here and that'LL leave us with a three ninety six to the fourth power And then one term that didn't cancel of is this one here So this also comes along So we haven't taken any limits yet. Just going ahead and simplifying as much as we can. So at this point, we have these products and the numerator and these last two terms are very similar. But notice that they're different This one and the top has an additional twenty six thousand. Well, I'm running out of room here so I'll go on to the next page picking up where I left off. Wait. Okay, so now we can go ahead and go ahead and from the previous page we can factor out at four So pull out a floor from the foreign plus four and then the remaining terms I'll leave them as they are And the reason I'm factoring out the first term is you'LL see in a second I'm going to be a pool So cancel with the term in the denominator Because before we had and plus one to the fourth Now we can go ahead and boobs we can go in and cancel one of those with this one appears, leaving us with the three and then we'll still have our three ninety six of the fourth and eleven o three twenty six thousand three ninety And so that was the only cancellation right there. And so now we're at this step Let's see what we can do next here. We want to cancel and we still have before here for in history. OK, so the next step I'LL go ahead and distributes through the parentheses here. So what I'll do is multiply this whole term out here through the Prentice is one will go to this and then the other one will go to this term here. So this isn't to be a bit more writing here. So and then the other one is where I distribute everything to this. So I'll just write that sermon first. I don't forget it and then I'll write the remaining foreign terms so foreign plus three foreign plus two foreign plus one So again all we did was we distributed this term here that our circle and blue we distributed this through the Prentice is here and you'LL see shortly why this is a good idea in the denominator we still have our same term Now if we go ahead and break up this into two fractions, we'll be able to cancel a bit. Oh, so are three ninety six of the fourth and plus one cube after cancellation. So we're breaking up. We're splitting this institute fractions. And in other words, we're going from something like this to this. And then I'm canceling after after work and similarly, the other fraction here we don't get to cancel like we did in the previous fraction and the previous fraction. We were able to cancel out thes two terms here, but not for the seconds room. But that's okay. So make note of this final expression here because I'll pick up where I left off in the next page. So next step I'LL do here in the next pages I'm gonna go ahead and factor out ofour from each of these. So when I do that end up with a total of two fifty six so that's four to the fourth because they have four fours and I'm multiplying there. And then after I take out the force, I have to factor it out from the parentheses. So here's my remaining expressions in the first fraction, and then for the remaining term down here, I'Ll just multiply that out and similarly do the same thing for the other fraction. So for the forest to fifty six and cute And then also after pulling out the four and then down here, I'LL pull out of And from this term and then I still have in Cube three and swear plus three and close one. But we also have Ah, we still have this three ninety six of the fourth we have to deal with too. Okay, so let's go on to the next step here of the simplification. So what? This step here, let me also I could also factor out and cube here, so and over here is well, so I will write Rewrite this It bit simplified here. So So remember the whole point of doing all this? Is that so? It's much easier for us to take a limit at the end. So we would still have this this end cubed up here. So I guess I'll write that right now, but you could see in a second old cancel and the denominator because if I pull out and cute here as well then in love with this And so here we can go ahead and cancel those that makes life a lot easier for us. And similarly over here. Second fraction You see that you have and Cube there, so you can factor out one here and cancel Yet here up that could rewrite a lot of this stuff out. So this will take a moment and then down here and from this I'LL leave this part alone. And then here I pull out and Hughes, just like previous step and then don't forget the three ninety six of the fourth. So now we could cancel this. And then Now we're at the point where we can go ahead and take a limit if we take a limit us and goes to infinity and see what we have left over. Well, for the first time up here, what we have as n goes to infinity, the error goes to zero zero zero. So we just have a one here with the two fifty six over three ninety six of the fourth. So brought the point. Now we're completing the ratio test and taking that limit, let me re write this out two, five, six over three, ninety six to the fourth. That's the first for action. And then plus, now we go to the second fraction. So the end's cancel These were all goto zero. So the apprentices are gotta one and then you have this on top. But then the denominator goes to infinity because of this and here. So if you want, you could informally right this as. And then this part will go to zero. And this is going to two, six, three, ninety times three, ninety six to the fourth. But the denominators infinitely large. So the fraction goes to zero. And that just means that we're left with the first fraction two fifty six, three, ninety six to the fourth, which is for over three ninety six of the fourth, which is definitely less than one so going on to the next page. So we conclude the Siri's converges by the ratio test and that results party now have a much shorter part be compared to this party for party. We'd like to know how many decimal places of pi do you get if you just take the first term of the Siri's or what if you did the first two terms, So let's go and answer that. So if we go ahead and take an equal zero corresponding to the first term Here. This would just be the zeros partial some, which is just the first term. And we just go to our formula for, and and plug in and equal zero into a in here and you're just left with Isn't this fraction which is two two o six route too. Nine eight o one. And remember, this is one over pi. So this is one over pi because the Siri's was given not his pie, but one over. Fine. So I should be an approximation. Sorry about that. So that means pies. Approximately nine, eight o one, two, two o six route, too. If we go to the calculator there, come and we see that we get an approximation that's accurate to six decimal places. Now it's going to the second part where we way use the first two terms. So the previous part was using one ceremony. Now we're going to. So this is where the first two terms will mean an equal zero and n equals one. So we have s one first partial some which is a not a one. And then you go ahead and plug in and equals one into the zero and one into the formula here and there we go. Go ahead and simplify this a little bit. Just gonna come and denominator up here. Um, one second. Let me see this. So this should be a before four nine out of nine for Let me go back and erase this. Okay, so then take the opposite of that the inverse, and get your pie. Remember, that's that's one over pi. So for pi, we get approximately nine. Eight o o one two root too. And then we flip this fraction here and then we're basically finished. Now, we would just go to the calculator, round this off and see how many decimals up I get. Two. So in this case, this one didn't give us much more this time will have fifteen places. So one, four, one, five, nine, two, six, five, three. And then we have five, eight, nine, and then seven, nine, three. And then after that, it breaks off. So this is correct, too. Fifteen places and that results party