GATE CS 2003 Q1 — Asymptotic Worst Case Time and Space Complexity

Consider the following $C$ function.

For large values of $y$ , the return value of the function $f$ best approximates

float f,(float x, int y) {
    float p, s; int i;
    for (s=1,p=1,i=1; i<y; i++) {
        p *= x/i;
        s += p;
    }
    return s;
}

View worked solution

Answer: B

To find what the function $f(x, y)$ approximates for large values of $y$ , let's trace the values of the variables $p$ and $s$ inside the loop.

Initially, before the loop starts:

The loop runs for $i = 1, 2, 3, \dots, y-1$ :

For $i = 1$ :
$p = p \times \frac{x}{1} = \frac{x}{1!}$
$s = s + p = 1 + \frac{x}{1!}$
For $i = 2$ :
$p = p \times \frac{x}{2} = \frac{x^2}{2!}$
$s = s + p = 1 + \frac{x}{1!} + \frac{x^2}{2!}$
For $i = 3$ :
$p = p \times \frac{x}{3} = \frac{x^3}{3!}$
$s = s + p = 1 + \frac{x}{1!} + \frac{x^2}{2!} + \frac{x^3}{3!}$

Continuing this process up to $i = y-1$ , the final value of $s$ returned by the function is:

s = 1 + \frac{x}{1!} + \frac{x^2}{2!} + \frac{x^3}{3!} + \dots + \frac{x^{y-1}}{(y-1)!}

For large values of $y$ (i.e., as $y \to \infty$ ), this infinite series is the standard Taylor series expansion of the exponential function:

e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!} = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \dots

Therefore, the return value of the function best approximates $e^x$ .

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Published 10 Oct 2026

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