The equivalent function 3

Computer Science Level 2

def fun(x,y):
    s,p=1,1
    for i in range(1,y):
        p*=x/float(i)
        s+=p
    return s

For sufficiently large value of $y$ , what does the above function closely approximate?

x^{y}

\sqrt{x}

e^{x}

\ln(x+1)

1 solution

Henry U
Dec 12, 2018

The for-loop that adds a number to $s$ in every iteration can be seen as a sum from 1 to $y$

$\displaystyle \sum_{i=1}^y \text{something}$

In every iteration, the previous value of $p$ is multiplied by $\frac x i$ , so the something that is added is a product of $i$ terms for every $i \leq y$ .

$\displaystyle \sum_{i=1}^y \prod_{j=1}^i \frac x j = \sum_{i=1}^y \frac {x^i} {i!}$

and these are exactly the first $y$ terms of the Taylor Series of the exponential function $\boxed{e^x}$ , so for sufficiently large $y$ , the sum converges to $e^x$ .

The equivalent function 3

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