First slide

Introduction to real valued functions

Question

Let $f (n) = 1 + \frac{1}{2} + \frac{1}{3} + \dots + \frac{1}{n}$ , then $(f (1) + f (2) + f (3) + \dots + f (n)$

Moderate

A

$n f (n) - 1$
B

$(n + 1) f (n) - n$
C

$(n + 1) f (n) + n$
D

$n f (n) + n$

Solution

We have $f (n) = 1 + \frac{1}{2} + \frac{1}{3} + \dots + \frac{1}{n}$
$\begin{array}{l} ∴ f (1) + f (2) + \dots + f (n) \\ = 1 + (1 + \frac{1}{2}) + (1 + \frac{1}{2} + \frac{1}{3}) + \dots + (1 + \frac{1}{2} + \frac{1}{3} + \dots + \frac{1}{n}) \\ = n + \frac{(n - 1)}{2} + \frac{(n - 2)}{3} + \dots + \frac{n - (n - 1)}{n} \\ = n (1 + \frac{1}{2} + \frac{1}{3} + \dots + \frac{1}{n}) - (\frac{1}{2} + \frac{2}{3} + \dots + \frac{n - 1}{n}) \\ = n (1 + \frac{1}{2} + \frac{1}{3} + \dots + \frac{1}{n}) - \{(1 - \frac{1}{2}) + (1 - \frac{1}{3}) + \dots + (1 - \frac{1}{n})\} \\ = n (1 + \frac{1}{2} + \frac{1}{3} + \dots + \frac{1}{n}) - \{(n - 1) - (\frac{1}{2} + \frac{1}{3} + \dots + \frac{1}{n})\} \\ = n f (n) - {(n - 1) - (f (n) - 1)} \\ = (n + 1) f (n) - n \end{array}$

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