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An infinite series pickle

One of the most important topics in your IB Math curriculum is sequences and series. You not only study how sequences of numbers can be related, but also how to apply such sequences in real world applications/patterns. More often than not, you'll see sequences and series as being part of another topic's exercise (in this case, trigonometry).

Question

[Maximum marks: 10]

Consider the following infinite sequence:

a.

i. Find the ratio of the sequence. [2 marks]

ii. Find the values of x for which the infinite series converges to a sum. [3 marks]

b. Find the sum of the series in terms of and . [1 mark]

c. Find the exact value(s) of for which the sum of the series is 2. [4 marks]

Solution

a. i. Given that

and that

we can evaluate the ratio using

ii. For an infinite series to converge to a sum, its ratio must be between -1 and 1 (not including the aforementioned values). That is,

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b. The infinite sum is given by

c. Given that the infinite sum must equal 2,

Squaring both sides of the equation,

Using the trigonometric identity

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Finally, applying the quadratic formula, you'll find

Given that, as stated previously, the value of needs to be between and , the only solution is