Infinite series

There are 4 NRICH Mathematical resources connected to Infinite series
Circles ad infinitum
problem
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Circles ad infinitum

Age
16 to 18
Challenge level
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A circle is inscribed in an equilateral triangle. Smaller circles touch it and the sides of the triangle, the process continuing indefinitely. What is the sum of the areas of all the circles?
Harmonically
problem
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Harmonically

Age
16 to 18
Challenge level
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Is it true that a large integer m can be taken such that: 1 + 1/2 + 1/3 + ... +1/m > 100 ?
Diminishing Returns
problem
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Diminishing Returns

Age
11 to 14
Challenge level
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How much of the square is coloured blue? How will the pattern continue?
Generally Geometric
problem

Generally Geometric

Age
16 to 18
Challenge level
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Generalise the sum of a GP by using derivatives to make the coefficients into powers of the natural numbers.