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There are **18** NRICH Mathematical resources connected to **Limits of sequences**, you may find related items under Patterns, sequences and structure.

Problem
Primary curriculum
Secondary curriculum
### Summing Geometric Progressions

Watch the video to see how to sum the sequence. Can you adapt the method to sum other sequences?

Age 14 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Diminishing Returns

How much of the square is coloured blue? How will the pattern continue?

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Litov's Mean Value Theorem

Start with two numbers and generate a sequence where the next number is the mean of the last two numbers...

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Squareness

The family of graphs of x^n + y^n =1 (for even n) includes the circle. Why do the graphs look more and more square as n increases?

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Slide

This function involves absolute values. To find the slope on the slide use different equations to define the function in different parts of its domain.

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Approximating Pi

By inscribing a circle in a square and then a square in a circle find an approximation to pi. By using a hexagon, can you improve on the approximation?

Age 14 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Climbing Powers

$2\wedge 3\wedge 4$ could be $(2^3)^4$ or $2^{(3^4)}$. Does it make any difference? For both definitions, which is bigger: $r\wedge r\wedge r\wedge r\dots$ where the powers of $r$ go on for ever, or $(r^r)^r$, where $r$ is $\sqrt{2}$?

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### A Swiss Sum

Can you use the given image to say something about the sum of an infinite series?

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### How Does Your Function Grow?

Compares the size of functions f(n) for large values of n.

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Light Blue - Dark Blue

Investigate the successive areas of light blue in these diagrams.

Age 7 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Ruler

The interval 0 - 1 is marked into halves, quarters, eighths ... etc. Vertical lines are drawn at these points, heights depending on positions. What happens as this process goes on indefinitely?

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Small Steps

Two problems about infinite processes where smaller and smaller steps are taken and you have to discover what happens in the limit.

Age 16 to 18

Challenge Level

Article
Primary curriculum
Secondary curriculum
### Infinite Continued Fractions

In this article we are going to look at infinite continued fractions - continued fractions that do not terminate.

Age 16 to 18

Article
Primary curriculum
Secondary curriculum
### Continued Fractions II

In this article we show that every whole number can be written as a continued fraction of the form k/(1+k/(1+k/...)).

Age 16 to 18

Article
Primary curriculum
Secondary curriculum
### Continued Fractions I

An article introducing continued fractions with some simple puzzles for the reader.

Age 14 to 18

Article
Primary curriculum
Secondary curriculum
### Zooming in on the Squares

Start with a large square, join the midpoints of its sides, you'll see four right angled triangles. Remove these triangles, a second square is left. Repeat the operation. What happens?

Age 7 to 14

Problem
Primary curriculum
Secondary curriculum
### Archimedes and Numerical Roots

The problem is how did Archimedes calculate the lengths of the sides of the polygons which needed him to be able to calculate square roots?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Little and Large

A point moves around inside a rectangle. What are the least and the greatest values of the sum of the squares of the distances from the vertices?

Age 16 to 18

Challenge Level