# `(asinalpha)/(1!) + (a^3sin^3alpha)/(3!) + (a^5sin^5alpha)/(5!) ` ...

electreto05 | Certified Educator

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)! + ...

neeldod | Student

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