Discrete Math Problem : Mathematical Induction

[GoGoDancer12]

Homework Statement

Prove that H₁ +H₂ +...+H_n = (n +1)(H_n-n)?

Homework Equations

H_n denotes the nth harmonic number.

The nth harmonic number is the sum of 1+1/2+...1/n,
which is n / n +1.

I'm not really sure if H_n = (1/ n) .

Prove by Mathematical Induction

H_n denotes the nth harmonic number.

The Attempt at a Solution

Basis Step : P(1) = H_n = (n +1)(H_n-n)
(1 / n) = (n +1) ( (1 / n) -n)
(1) = ( 1 + 1) (1-1)
1 = (2) (0)
1 = 0, which is false

or
Basis Step:
P(1) = Hn = H_n = (n +1)(H_n-n)
( n / n + 1) = (n + 1) ( (n / n + 1) -1)

P(1) is false, because 1/ 2 does not equal to - 1

 

[jbunniii]

Homework Statement

Prove that H₁ +H₂ +...+H_n = (n +1)(H_n-n)?

Homework Equations

H_n denotes the nth harmonic number.

The nth harmonic number is the sum of 1+1/2+...1/n,
which is n / n +1.

I'm not really sure if H_n = (1/ n) .

Well, it can't be both $n/(n+1)$ and $1/n$. In fact, I don't think it is either one. As far as I know, there is no simple formula for $H_n$.

Let's start with the induction hypothesis. Namely, we need to check that the formula is correct for $n = 1$. Plugging in $n = 1$, the formula reduces to

$$H_1 = (1 + 1)(H_1 - 1)$$

Is this correct?

 

[GoGoDancer12]

How would I know if its correct or not if there is no simple formula for H_n ??

 

[jbunniii]

How would I know if its correct or not if there is no simple formula for H_n ??

There may not be a simple formula for $H_n$ in general, but what's $H_1$? Isn't it simply 1? (Look at the defining sum.)

 

[GoGoDancer12]

I think H₁= (1 / n) = 1

 

[jbunniii]

I think H₁= (1 / n) = 1

Right, so is the formula correct for $n = 1$? i.e. is it true that

$$H_1 = (1+1)(H_1 - 1)$$

If not, then there is something wrong with the problem statement.

 

[GoGoDancer12]

No, its not true because H_n = 1
and (1+1)(1-1) = 0

So, there's no way to prove H₁ +H₂+...+H_n = (n +1) (H_n - n)
is true, right??

 

[jbunniii]

No, its not true because H_n = 1
and (1+1)(1-1) = 0

So, there's no way to prove H₁ +H₂+...+H_n = (n +1) (H_n - n)
is true, right??

Correct. If it isn't even true for $n = 1$ then it's certainly not a valid formula.