Question 0517

Sigma Notation

2017 Paper 1 Question 9 Variant

Question

(a)

A sequence

{u_1, u_2, u_3 \ldots}

has a sum

{S_n}

where

{S_n = \displaystyle \sum_{r=1}^n u_r.}

It is given that

{S_n = A n^2 + B n,}

where

{A}

and

{B}

are non-zero constants.

(ai)

Find an expression for

{u_n}

in terms of

{A, B}

and

{n.}

Simplify your answer.

[3]

(aii)

It is also given that the fourteenth term is

{190}

and the twenty-first term is

{288.}

Find

{A}

and

{B.}

[2]

(b)

Show that

r(r+1)(2r+1)-(r-1)r(2r-1) = kr^2,

where

{k}

is a constant to be determined.

Use this result to find a simplified expression for

\sum_{r=1}^n r^{2}.

[4]

(c)

D'Alembert's ratio test states that a series of the form

\sum_{r=0}^\infty a_r

converges when

\lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| < 1,

and diverges when

\lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| > 1.

When

\lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| = 1,

the test is inconclusive.

Using the test, explain why the series

\sum_{r=0}^\infty \frac{(-1)^r x^{2r}}{(2r)!}

converges for all real values of

{x}

and state the sum to infinity of this series, in terms of

{x.}

[4]