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Proof Using General principle of math induction

  • Thread starter kolley
  • Start date
  • #1
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Homework Statement



prove that 1+1/4+1/9+...+1/n^2< or = 2-1/n for every positive integer n

Homework Equations





The Attempt at a Solution



proved it was correct for n=1, then replaced the n with k, changed it to k+1 to get:

1/(k+1)^2 < or = 2-1/(k+1)

don't know how to proceed

Homework Statement





Homework Equations





The Attempt at a Solution

 

Answers and Replies

  • #2
33,173
4,858

Homework Statement



prove that 1+1/4+1/9+...+1/n^2< or = 2-1/n for every positive integer n

Homework Equations





The Attempt at a Solution



proved it was correct for n=1, then replaced the n with k, changed it to k+1 to get:

1/(k+1)^2 < or = 2-1/(k+1)

don't know how to proceed
Note: <= means "less than or equal to." You don't need to write < or =.
You're not supposed to show that 1/(k+1)^2 <= 2-1/(k+1). You need to show that 1+1/4+1/9+...+1/(k + 1)^2 <= 2 - 1/(k + 1).

Your induction hypothesis is 1+1/4+1/9+...+1/k^2 <= 2 - 1/k. How can you get from this statement to the one you want to prove?
 
  • #3
17
0
Sorry, I left out part of mine. I had 49/36 +1/(k+1)^2 <= 2-1/(k+1)

since 1+1/4+1/9 is equal to 49/36, is this correct or am I still on the wrong track?
 
  • #4
16
0
Your part where u show the base case is correct but u can't just replace k with k+1. otherwise it would be a tautology not a proof. you have to show you can get it into the form where 1+1/4+1/9+...+1/k+1/(k+1) <= 2 - 1/(k+1)

Here is a simple proof: Show by induction that

[tex]
1+2+3+...+n = \frac{n*(n+1)}{2}
for n = 1 1*(1+1) = \frac{2}{2} = 1. so this is true for the base case.
[/tex]

now using rules of algebra if we add to one side, we add to the other, so

[tex]
1+2+3+..+n+(n+1) = \frac{n*(n+1)}{2} + (n+1) = \frac{n^2+3n+2}{2} = \frac{(n+1)*(n+2)}{2}= \frac{(n+1)*((n+1)+1)}{2}
[/tex]

So by induction, we are done.
 
Last edited:
  • #5
33,173
4,858
Sorry, I left out part of mine. I had 49/36 +1/(k+1)^2 <= 2-1/(k+1)

since 1+1/4+1/9 is equal to 49/36, is this correct or am I still on the wrong track?
You're still on the wrong track.

1+1/4+1/9+...+1/n^2 does not mean 1 + 1/4 + 1/9 + 1/n^2. The ellipsis - the three dots -- means "continuing in the same fashion." IOW, it means 1 + 1/4 + 1/9 + 1/16 + 1/25 + ... and so on, up to 1/n^2 for whatever value n happens to be.
 

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