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“Summing Series: Advanced Techniques” is available to Standard subscribers.

The series that JEE, A-Level Further Maths and IB Higher Level ask most: sums of n, n squared and n cubed, arithmetico-geometric series, telescoping sums and the method of differences, with a full timed paper.

3Lessons11Problems2Python Labs1Question Set1Mock Test

Read the first lesson freeSums of powers of natural numbers

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Sums of powers of natural numbers: The sum of the first $n$ natural numbers is $\dfrac{n(n + 1)}{2}$ , of their squares $\dfrac{n(n + 1)(2n + 1)}{6}$

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Arithmetico-geometric series: An arithmetico-geometric series multiplies an AP by a GP: its terms are

[a + (k - 1)d]\,r^{k - 1}

. To add

n

terms, write

S

and

rS

one under the other, shifted by one place, and subtract: a geometric series is left in the middle. For

|r| < 1

the sum to infinity is

\dfrac{a}{1 - r} + \dfrac{dr}{(1 - r)^2}

Telescoping and the method of differences: A telescoping sum has terms of the form

f(k) - f(k + 1)

, so everything cancels except

f(1) - f(n + 1)

. Partial fractions turn terms like

\dfrac{1}{k(k + 1)}

into such differences. When the gaps between the terms of a series form an AP or a GP, find the

n

th term from the gaps first. A harmonic progression has reciprocals in AP.