Math  /  Algebra

QuestionFind the sum of the series S=i=1x4i,x<1S=\sum_{i=1}^{\infty} x^{4 i},|x|<1 S=S=

Studdy Solution

STEP 1

What is this asking? We need to find the sum of an infinite geometric series with powers of x4x^4. Watch out! Remember the formula for the sum of an infinite geometric series only works when the absolute value of the common ratio is less than 1, which is thankfully given here!

STEP 2

1. Rewrite the series
2. Identify the first term and common ratio
3. Apply the infinite geometric series formula

STEP 3

Let's write out the first few terms of our series S=i=1x4iS = \sum_{i=1}^{\infty} x^{4i} to get a better feel for it.
We have S=x4+x8+x12+x16+S = x^4 + x^8 + x^{12} + x^{16} + \dots

STEP 4

Look closely!
This is a geometric series.
Each term is the previous term multiplied by x4x^4.

STEP 5

The **first term**, when i=1i=1, is a=x4a = x^4.

STEP 6

The **common ratio** is r=x4r = x^4.
We multiply each term by x4x^4 to get the next term.

STEP 7

Remember, the sum of an infinite geometric series is given by the formula a1r\frac{a}{1 - r} where aa is the **first term** and rr is the **common ratio**, as long as r<1|r| < 1.
They told us that x<1|x| < 1, so we know x4<1|x^4| < 1 too, which is great!

STEP 8

Let's **plug in** our values: a=x4a = x^4 and r=x4r = x^4. x41x4\frac{x^4}{1 - x^4} This is our **sum**!

STEP 9

The sum of the series is x41x4\frac{x^4}{1 - x^4}.

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