Let n ? 1 be a integer ( ie. n ? ? ) and defines the sum Sn...

Let n ? 1 be a integer ( ie. n ? ? ) and defines the sum S_n = 1/(1 · 2) + 1/(2 · 3) + 1/(3 · 4) + ··· + 1/(n(n + 1)) calculate S_n = 1, 2, 3, ..., enough to guess a formula for S_n. Then prove the formula using mathematical induction

Transcript

00:01 Okay, we want to prove that i times i plus 1 when i goes from 1 to n, if you add all those up, you get n times n plus 1 times n plus 2 all over 3.

00:12 Okay, oh, sorry, for n greater than or equal to 1.

00:18 Okay, we're going to do this proof by induction.

00:21 That means we're going to first show that it works for n equals 1.

00:25 Then we're going to assume that it works for n equals k, and then we're going to prove that it works for n equals k plus 1.

00:32 Okay, so step one.

00:34 Show the statement is true for n equals 1.

00:48 So i equals 1 to 1, i times i plus 1 equals 1 times 1 plus 1, which equals 2.

00:59 And then the other side is 1 times 1 plus 1 times 1 plus 2.

01:06 Oops one plus two all over three equals one times two times three all over three equals two so i equals one to one i i i plus one equals one one plus one plus two all over three okay so it's true for i equals one second assume it is true for n equals k that is always write this down what is it that you're assuming well you're assuming that i equals 1 to k i times i plus 1 equals k k plus 1 k plus 2 all over 3 okay now show it is true for n equals k plus 1.

02:21 That is show i equals 1 to k plus 1 i times i plus 1 equals k plus 1 k plus 1 plus 1 plus 2 all over 3.

02:50 I'm going to go ahead and simplify this a little bit it's k plus 1 k plus 2 k plus 3 all over 3.

03:01 Okay so now you're going to to do the math.

03:06 So up here when you're showing it was true for n equals one, you did math here and you did math here.

03:13 Okay, now here, i'm telling you what i'm getting ready to do and now i'm going to get ready to do the math.

03:19 And the math is, just like a trig proof.

03:21 I'm going to start here on the left.

03:23 I'm going to do some substitutions and manipulations and stuff and i'm going to end up with this here on the right.

03:29 Once i get there, then i know that i'm done.

03:33 Okay, so i'm going to start with this thing right here and what i'm going to do is i'm going to break it down into i equals 1 to k plus the extra term.

03:42 The reason i'm going to do that is because once i get the i equals 1 to k which is right here i can plug this in for it.

03:52 All right so here i go...

Question

Let n ? 1 be a integer ( ie. n ? ? ) and defines the sum S_n = 1/(1 · 2) + 1/(2 · 3) + 1/(3 · 4) + ··· + 1/(n(n + 1)) calculate S_n = 1, 2, 3, ..., enough to guess a formula for S_n. Then prove the formula using mathematical induction

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