Question
Mon June 11, 2012

# Using principle of mathematical induction, prove that 1+1/4+1/9+1/16+...+1/n square 2 and n=2.

Expert Reply
Tue June 12, 2012

When n = 2, we have,

1 + 1/4 < 2 - 1/2

5/4 < 3/2, which is a true statement.

We now need to show that if:

1 + 1/4 + 1/9 + 1/6 + ... + 1/n2 < 2 - 1/n.

Then: 1 + 1/4 + 1/9 + 1/6 + ... + 1/n2 + 1/(n + 1)2 < 2 - 1/(n + 1).

We have:
1 + 1/4 + 1/9 + 1/6 + ... + 1/n2 + 1/(n + 1)2
< 2 - 1/n  + 1/(n + 1)2, by the Induction hypothesis
= 2 - (n2 + n + 1)/[n(n + 1)2]

From here, we just need to show that:
2 - (n2 + n + 1)/[n(n + 1)2] < 2 - 1/(n + 1) for all n ? 2.

Solving this inequality:

2 - (n2 + n + 1)/[n(n + 1)2] < 2 - 1/(n + 1)
=> -(n2 + n + 1)/[n(n + 1)2] < -1/(n + 1)
=> (n2 + n + 1)/[n(n + 1)2] > 1/(n + 1)
=> (n2 + n + 1)/[n(n + 1)2] - 1/(n + 1) > 0
=> [(n2 + n + 1) - (n2 + n)]/[n(n + 1)2] > 0
=> 1/[n(n + 1)2] > 0.

Since this holds for n > 0. We have shown that:
2 - (n2 + n + 1)/[n(n + 1)2] < 2 - 1/(n + 1) for all n ? 2.

Thus, (1 + 1/4 + 1/9 + 1/6 + ... + 1/n2) + 1/(n + 1)2 < 2 - 1/(n + 1).

Thus, by Principle of Mathematical induction, the result is true for all n greater than equal to 2.

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