Convergence and Limits of Sequences

A question is this type if and only if it requires determining whether a sequence converges, finding its limit, or analyzing convergence conditions for sequences defined by recurrence relations.

5 questions · Challenging +1.1

8.01a Recurrence relations: general sequences, closed form and recurrence

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OCR Further Additional Pure AS 2019 June Q2

4 marks Challenging +1.2

The convergent sequence $\left\{ \mathrm { a } _ { \mathrm { n } } \right\}$ is defined by $a _ { 0 } = 1$ and $\mathrm { a } _ { \mathrm { n } + 1 } = \sqrt { \mathrm { a } _ { \mathrm { n } } } + \frac { 4 } { \sqrt { \mathrm { a } _ { \mathrm { n } } } }$ for $n \geqslant 0$. Calculate the limit of the sequence.
The convergent sequence $\left\{ \mathrm { b } _ { \mathrm { n } } \right\}$ is defined by $\mathrm { b } _ { 0 } = 1$ and $\mathrm { b } _ { \mathrm { n } + 1 } = \sqrt { \mathrm { b } _ { \mathrm { n } } } + \frac { \mathrm { k } } { \sqrt { \mathrm { b } _ { \mathrm { n } } } }$ for $n \geqslant 0$, where $k$ is a constant. Determine the value of $k$ for which the limit of the sequence is 9 .

OCR Further Additional Pure AS Specimen Q1

3 marks Standard +0.3

1 The sequence $\left\{ u _ { n } \right\}$ is defined by $u _ { 1 } = 2$ and $u _ { n + 1 } = \frac { 12 } { 1 + u _ { n } }$ for $n \geq 1$.
Given that the sequence converges, with limit $\alpha$, determine the value of $\alpha$.

OCR Further Additional Pure 2023 June Q4

7 marks Hard +2.3

4 The sequence $\left\{ A _ { n } \right\}$ is given for all integers $n \geqslant 0$ by $A _ { n } = \frac { I _ { n + 2 } } { I _ { n } }$, where $I _ { n } = \int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \cos ^ { n } x d x$.

Show that $\left\{ A _ { n } \right\}$ increases monotonically.
Show that $\left\{ \mathrm { A } _ { \mathrm { n } } \right\}$ converges to a limit, $A$, whose exact value should be stated.

OCR Further Additional Pure 2020 November Q8

12 marks Challenging +1.2

8 The sequence $\left\{ u _ { n } \right\}$ of positive real numbers is defined by $u _ { 1 } = 1$ and $u _ { n + 1 } = \frac { 2 u _ { n } + 3 } { u _ { n } + 2 }$ for $n \geqslant 1$.

Prove by induction that $u _ { n } ^ { 2 } - 3 < 0$ for all positive integers $n$.
By considering $u _ { n + 1 } - u _ { n }$, use the result of part (a) to show that $u _ { n + 1 } > u _ { n }$ for all positive integers $n$. The sequence $\left\{ u _ { n } \right\}$ has a limit for $n \rightarrow \infty$.
Find the limit of the sequence $\left\{ u _ { n } \right\}$ as $n \rightarrow \infty$.
Describe as fully as possible the behaviour of the sequence $\left\{ u _ { n } \right\}$. \section*{END OF QUESTION PAPER}

Edexcel C1 Q38

7 marks Standard +0.3

A sequence is defined by the recurrence relation $$u_{n+1} = \sqrt{\frac{u_n}{2} + \frac{a}{u_n}}, \quad n = 1, 2, 3, \ldots,$$ where $a$ is a constant.

Given that $a = 20$ and $u_1 = 3$, find the values of $u_2$, $u_3$ and $u_4$, giving your answers to 2 decimal places. [3]
Given instead that $u_1 = u_2 = 3$,
1. calculate the value of $a$, [3]
2. write down the value of $u_5$. [1]