Induction Inequality Proof Example 1: Σ(k = 1 to n) 1/k² ≤ 2 - 1/n

Induction proofs involving sigma notation look intimidating, but they are no more difficult than any of the other proofs that we've encountered!

[blazept]

lol i can truly say this is a coincidence: the reason why i started watching your videos was because i was struggling with this EXACT inequality which was assigned for my homework! Thank you very much for your help!

[ievgenieris]

This is the best lesson for induction I've seen even though it is just about a single example

[MrAlex]

I think you have a mistake at 6:29, when you expand the fractions. Shouldn't it be 2 - 1/c + 1/(c+1)^2 = 2 + (-1)/c + 1/(c+1)^2 = 2 + ((-(c+1)^2) + c)/(c(c+1)^2) = 2 - ((c+1)^2 - c)/(c(c+1)^2) ?

[namrashah]

I wasn't really able to do this by using (c+1)² - c instead of (c+1)² + c at 6:30 . Just changes the solution .

[LoLzWatsUsay]

At around 8:31 why are the terms subtracted instead?
All you did was split the fraction up, so why does it have to be subtracted?

[asimami]

How you make a + to - in this question can you please explain.

[AwkwardFX]

Induction Inequality Proof Example 1: Σ(k = 1 to n) 1/k² ≤ 2 - 1/n 

At time 5:44, how did you conjure up 2- (1/c+1) ? I mean, why do we have to specifically put it in that form? And how did you derive that expression? 

[TheJProductins]

Thank you!! The algebra was so mind bending on this one. I gave up after a while. Thank you for the video!

[timmoneel]

Thank you for your proof! I am trying to help one of my family members to understand proof by induction. Your method was easy to follow. Thanks again!

[syedalihasan]

I am stuck in a question of similar nature, could you please work it out for me..
1/1^1/2 + 1/2^1/2+...+ 1/n^1/2 > or equal to (n+1)^1/2

[bradypinter]

Jesus Christ this man just saved my life

[rosjohn]

At 8:43, why did you change the fractions to subtraction instead of addition?

[RathikMurtinty]

For those saying this proof is incorrect, change the sign of c at 6:26 and work the proof out. The numerator will be c^2 + 2c + 1 - c which simplifies to c^2 + c + 1 = c(c+1) + 1. If you continue with the steps after you will get the same answer.

[simon]

This is really complicated! You can use a telescoping sum: Sum(k=1, n) 1/k^2 <= Sum(k=1, n) 1/(k*(k-1)) = 1+Sum(k=2, n)1/(k*(k+1)) = 1+ Sum(k=2, n) [ 1/k-1 - 1/k ] =(here comes the telescoping sum) 1+ 1 - 1/n = 2 -1/n <= 2. But if you really want to use Induction...

[svetlanatemesheva]

06:23, error, must be 2-\frac{(c+1)^2-c}{c(c+1)^2)}

[tharealminipunch]

Thanks for the help! Better than my professor for sure!

[abobrhom]

Why did you change the positive sign to negative sign ?? I think that is wrong !

[reubenmanzo]

In the 'Assume true for n=c' line, isn't it already given that c must be a positive integer, since c can only take the numbers that n can take?

[isaacstrid]

I like it second last line 3 <= 5 😂cracked me up great vid eddie

[eftie]

Sorry but why its minus? when you change it from a plus to a minus you have to change < to > and than its wrong