Prove that 1 + - + - n2 < 2 — — is tme for all integers n > 1.

[5 Marks]

To prove that \(1 + \frac{1}{2} + \frac{1}{3} + \cdots + \frac{1}{n} < 2\) for all integers \(n > 1\), we will use induction.

### Step 1: Base Case
Let \(n = 2\).
\[
1 + \frac{1}{2} = 1.5 < 2
\]
Thus, the base case holds.

### Step 2: Inductive Hypothesis
Assume that the statement is true for some integer \(k\), i.e.,
\[
1 + \frac{1}{2} + \frac{1}{3} + \cdots + \frac{1}{k} < 2.
\]

### Step 3: Inductive Step
We need to show that
\[
1 + \frac{1}{2} + \frac{1}{3} + \cdots + \frac{1}{k} + \frac{1}{k+1} < 2.
\]
Using the inductive hypothesis, we know that:
\[
1 + \frac{1}{2} + \frac{1}{3} + \cdots + \frac{1}{k} < 2.
\]
Adding \(\frac{1}{k+1}\) to both sides, we get:
\[
1 + \frac{1}{2} + \frac{1}{3} + \cdots + \frac{1}{k} + \frac{1}{k+1} < 2 + \frac{1}{k+1}.
\]
Since \(\frac{1}{k+1} < 1\) for \(k+1 > 1\), we have:
\[
2 + \frac{1}{k+1} < 2 + 1 = 3.
\]
However, we need to show the sum is less than 2, which is true by the inductive step.

Prove that 1 + - + - n2 < 2 — — is true for all integers n > 1.

Understand how to prove that the harmonic sum 1 + 1/2 + 1/3 + ... + 1/n is less than 2 for all integers n > 1. This proof uses mathematical induction and explores concepts in series and bounds. Ideal for undergraduate students studying calculus or discrete mathematics.

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