Infinite Series Formula

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Infinite Series Formula

The infinite series formula is used to find the sum of a sequence where the number of terms is infinite. There are various types of infinite series. In this section, we will discuss the sum of infinite arithmetic series and the sum of infinite geometric series. The arithmetic series is the sequence where the difference between each consecutive term is constant throughout and the geometric series is the series where the ratio of the consecutive terms to the preceding term is the same throughout. The infinite series formula is a handy tool to calculate the sum very quickly. Let us learn more about the infinite series formula along with solved examples.

What Is Infinite Series Formula?

The sum of the infinite geometric series formula is used to find the sum of the series that extends up to infinity. This is also known as the sum of infinite GP. While finding the sum of a GP,  we find that the sum converges to a value, though the series has infinite terms. The infinite series formula if −1<r<1, can be given as,

  • Sum = a/(1-r)

Where,

  • a = first term of the series
  • r = common ratio between two consecutive terms and −1 < r < 1

Note: If r > 1, the sum does not exist as the sum does not converge.

  • The sum of an infinite arithmetic sequence is ∞, if d > 0, or
  • The sum of an infinite arithmetic sequence is ∞, if d > 0- ∞, if d < 0.

Let us now have a look at a few solved examples using the Infinite Series Formula.

Examples using Infinite Series Formula

Example 1: Using an infinite series formula, find the sum of infinite series: 1/4 + 1/16 + 1/64 + 1/256 +⋯

Solution:

Given: a = ¼

 r = (1/16) / (1/4) = (1/64) / (1/16) = ¼

To find: Sum of the given infinite series

If r<1 is then sum is given as Sum = a/(1-r)

Applying the values to the infinite series formula, we get

Sum=(1⁄4)/(1-1⁄4)

Sum=(1⁄4)/(3⁄4)

Sum=4/(3*4)

Sum=1/3

Answer: The sum of 1/4+1/16+1/64+1/256+⋯ is 1/3

Example 2: Using the infinite series formula, find the sum of infinite series: 1/2 + 1/6 + 1/18 + 1/54 + ⋯

Solution:

Given: a = 1/2

 r = (1/6) / (1/2) = (1/18) / (1/6) = 1/3

To find: Sum of the given infinite series

If r<1 is then sum is given as Sum = a/(1-r)
Applying the values to the infinite series formula, we get

Sum=(1⁄2)/(1-1⁄3)

Sum=(1⁄2)/(2⁄3)

Sum=3/(2*2)

Sum=3/4

Answer: The sum of 1/2 + 1/6 + 1/18 + 1/54 + ⋯ is 3/4

Example 3: Evaluate 3 + 7 + 11 + …….
Solution:

a = 3, d = 4 and n = ∞

Here the difference > 0.
So, the sum = + ∞

Answer: 3 + 7 + 11 + …….  = + ∞

FAQs on Infinite Series Formula

What Is the Sum of Infinite Terms?

An infinite series has an infinite number of terms. The sum of the first n terms, Sn, is called a partial sum. If Sn tends to a limit as n tends to infinity, the limit is called the sum to infinity of the series. The sum of infinite arithmetic series is either +∞ or – ∞. The sum of the infinite geometric series when the common ratio is <1, then the sum converges to a/(1-r), which is the infinite series formula of an infinite GP. Here a is the first term and r is the common ratio.

What Is the Infinite Series Formula?

The sum of infinite arithmetic series is either +∞ or – ∞. The sum of the infinite geometric series formula is also known as the sum of infinite GP. The infinite series formula if the value of  r is such that −1<r<1, can be given as,

Sum = a/(1-r)

Where,

  • a = first term of the series
  • r = common ratio between two consecutive terms and −1<r<1

What Is a and r in Infinite Series Formula?

In finding the sum of the given infinite geometric series If r<1 is then sum is given as Sum = a/(1-r). In this infinite series formula, a = first term of the series and r = common ratio between two consecutive terms and −1<r<1.

Find the sum of the infinite GP 0.3+ 0.33+ 0.333+….

This infinite GP can be written as, 3/10 + 3/100 + 3/1000+……….

Here we find that the first term a = 3/10, and r = 3/100 ÷ 3/10 = 1/ 10

Since r < 1, the sum must converge to a/ (1-r) as per the infinite series formula for infinite GP.

Thus the sum to infinity = (3/10) ÷ (1 – 1/10)

Sum = 3/10 ÷ 9/10 = 1/3

Thus the sum converges to 1/3.  0.3+ 0.33+ 0.333+…. = 1/3

Infinite Series

An infinite series has an infinite number of terms.
The sum of the first n terms, Sn , is called a partial sum.

If Sn tends to  a limit as n tends to infinity, the limit is called the sum to infinity of the series.

Arithmetic series

Example

Express the recurring decimal  0.242424…. as a vulgar fraction

Example

Given that 12 and 6 are two adjacent terms of an infinite
geometric series with a sum to infinity of 192,

a) Find the first term.
b) Find the partial sum S6

Solution
a)

b)

Sigma notation – Rules

Adding series

Sum of first n natural numbers

Useful result

Example

Find the value of

From the rules above,

Common series  – Sigma notation

Power Series

Absolute convergence

If

Example

Find the sum to infinity of the series

4 + 7x +10x2+13x3+…

and the region of valid values of x.

Fibonacci series

Centre of convergence

The power series can be written

where c is the centre of convergence.
This is the middle of the interval of convergence,
the interval for which the limit exists.
The radius of convergence is called R.

If R = ∞ , the series converges for all x.
Otherwise,

go again

and

Substituting back into the original series

In the interval (-R+c,R+c)

Taylor’s series

Infinite Series

The sum of infinite terms that follow a rule.

When we have an infinite sequence of values:

First Example

You might think it is impossible to work out the answer, but sometimes it can be done!

Using the example from above:

Notation

We often use Sigma Notation for infinite series. Our example from above looks like:

Converge

Let’s add the terms one at a time. When the “sum so far” approaches a finite value, the series is said to be “convergent“:

Our first example:

“the sequence of partial sums has a finite limit.”

Diverge

If the sums do not converge, the series is said to diverge.

It can go to +infinity, −infinity or just go up and down without settling on any value.

Example: 1 − 1 + 1 − 1 + 1 …

It goes up and down without settling towards some value, so it is divergent.

More Examples

When the difference between each term and the next is a constant, it is called an arithmetic series.

Harmonic Series

This is the Harmonic Series:

It is divergent.

Here is another way:

We can sketch the area of each term and compare it to the area under the 1/x curve:

Calculus tells us the area under 1/x (from 1 onwards) approaches infinity, and the harmonic series is greater than that, so it must be divergent.

Alternating Series

An Alternating Series has terms that alternate between positive and negative.

It may or may not converge.

Another example of an Alternating Series (based on the Harmonic Series above):

This one converges on the natural logarithm of 2

You can investigate this further!

  • do those rectangles really make the area above the curve as shown?
  • is the area below the curve really ln(2) = 0.693… ? Try the Integral Approximation Calculator to see (y=1/x between 1 and 2)

Order!

The order of the terms can be very important! We can sometimes get weird results when we change their order.

For example in an alternating series, what if we made all positive terms come first? So be careful!

The sum of an infinite series

Math Formulas ⭐️⭐️⭐️⭐️⭐

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