It seems that aims to determine the general rule or general term of this succession. There is no calculation method for this operation; the procedure involves the observation of the behavior of its elements.
In this case we see the behavior of the exponents and the factorial value of each element; then this behavior is expressed as a function of a variable n, evaluated, for the set of natural numbers.
We have the following succession:
a sin α/1! + a^3 sin^3 α/3! + a^5 sin^5 α/5! + ….. + a^k sin^k α/k!
Where k expresses the rule to determine. In this case we see that k meets the following expression for each term:
k = 2n – 1; for n = 1, 2, 3, 4, .....
For n = 1; k = 2(1) – 1 = 1
For n = 2; k = 2(2) – 1 = 3
For n = 3; k = 2(3) – 1 = 5
So the expression that generates the terms of this succession can be written as:
a^(2n – 1) sin^(2n – 1) α/(2n – 1)!
Then we can write the succession as follows:
a sin α/1! + a^3 sin^3 α/3! + a^5 sin^5 α/5! + ...
... + a^(2n – 1) sin^(2n – 1) α/(2n – 1)! + ...
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