JEE Mains Maths · Definite Integration

Limits of Sums as Integrals

Limits of long sums rewritten as (1/n) Σ f(k/n) and evaluated as a definite integral, with the limits set by the range of k.

All Definite Integration notes JEE Mains Maths strategy

Why this matters

Seven PYQs, the smallest page in the chapter, and all the same move: divide by the right power of n until each term reads (1/n) f(k/n), then integrate. The only choices are f and the limits. One idea covers the page.

Concept 1 of 1: Sums as integrals

\lim_{n\to\infty}\frac1n\sum_{k=1}^nf\left(\frac kn\right)=\int_0^1f(x)\,dx

: the sum adds strips of width

\frac1n

and height

f\left(\frac kn\right)

. Divide the numerator and denominator of each term by the power of

n

that leaves a factor

\frac1n

times a function of

\frac kn

. If

k

runs to

2n

the upper limit is 2; constants outside the sum stay outside.

Definition

$\lim\frac1n\sum_{k=1}^{n}f\left(\frac kn\right)=\int_0^1f$ .
$k$ up to $pn$ : $\int_0^pf$ .
Replace $\frac kn$ by $x$ and $\frac1n$ by $dx$ .

Riemann sum

\lim_{n\to\infty}\frac1n\sum_{k=1}^{n}f\!\left(\frac kn\right)=\int_0^1 f(x)\,dx

Worked example

Find

\lim_{n\to\infty}\sum_{k=1}^n\frac{n}{n^2+k^2}

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

JEE Mains · 2022 · 14 June 2022 · Q157Moderate

Example 1 · Definite Integration · Limits of Sums as Integrals

\lim_{n \rightarrow \infty} \left( \frac{n^{2}}{\left( n^{2}+ 1 \right)(n + 1)}+\frac{n^{2}}{\left( n^{2}+ 4 \right)(n + 2)}+\frac{n^{2}}{\left( n^{2}+ 9 \right)(n + 3)}+ \ldots +\frac{n^{2}}{\left( n^{2}+n^{2} \right)(n + n)} \right)

is equal to

Count the terms

The range of

k

sets the upper limit. A sum to

2n

3n

integrates to 2 or 3, not 1.

Drill 6 more on sums as integrals

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (1)

Sums as integrals
Riemann sum
$\lim_{n\to\infty}\frac1n\sum_{k=1}^{n}f\!\left(\frac kn\right)=\int_0^1 f(x)\,dx$

Watch out for (1)

Count the terms
→ Sums as integrals

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