1) Let P(n) be .... 2) Base Case (for n=1 on this set of positive integers) in w

Question

1) Let P(n) be ....

2) Base Case (for n=1 on this set of positive integers) in which you can plug in n=1 to both sides of the inequality and get the same answer.

3) Induction step.

I asked this question before, including what I thought to be the correct function for P(n), but did not get an answer that worked out. That is why I am asking specifically for what you consider P(n). Before, I tried to use summation equations for 1/n and 1/(2^n) and neither worked with the inequality above, mainly because of the 1/3 in the left-hand side vs the last term on the LHS. I would really appreciate your expertise in setting up this proof correctly.

The link to one previous attempt is here if that will help clarify any questions about what I did wrong the first time: http://www.cramster.com/answers-mar-12/advanced-math/proof-induction-summatio-textbook-mathematics-discrete-introduct_2233001.aspx?rec=0

Explanation / Answer

let P(n) be the inequality 1 + 1/2 + 1/3 +1/4 +.......... +1/2n >= 1 +n/2   where n is a positive integer

Base case:

when n=1 ;

P(1) implies  1 +1/2 >= 1+ 1/2  (true)

Induction step :

let P(n) be true 

i.e , 1 + 1/2 + 1/3 +1/4 +.......... +1/2n >= 1 +n/2     ----------(1)

Now , consider    1/(2n+1) >= 1/2n+1+

                     1/(2n+2) >= 1/2n+1

                     1/(2n+3) >= 1/2n+1  

                    1/(2n+4) >= 1/2n+1  and so on...

                    1/(2n +2n) >= 1/2n+1

----------------------------------------------------------------------------

adding all these , we get     1/(2n+1) +1/(2n+2) +1/(2n+3) +1/(2n+4) +................+1/(2n+2n) 

                                     >= 1/2n+1 + 1/2n+1 + 1/2n+1 + 1/2n+1 + .......................+ 1/2n+1  (2n terms are present)

                                   >=  2n / 2n+1

                                   >= 1/2

Therefore ,    1/(2n+1) +1/(2n+2) +1/(2n+3) +1/(2n+4) +................+1/(2n+2n)  >= 1/2  ------------(2)

Adding (1) and (2) , we get

    1 + 1/2 + 1/3 +1/4 +.......... +1/2n +  1/(2n+1) +1/(2n+2) +1/(2n+3) +1/(2n+4)+................+1/(2n+2n)                                                                  

                                                                        >= 1 +n+1/2 

1.e , 

    1 + 1/2 + 1/3 +1/4 + ....................... + 1/2n+1 > =  1+ (n+1)/2

P(n+1) is true

Therefore , P(n) is true by induction   :)

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