Infinite Series Formula, Relationship b/w A.M. and G.M, Sample Questions

Infinite Series Formula helps to calculate the sum of a sequence where the number of terms is infinite. Infinite series is the series that contains infinite terms and Sn which is the sum of the first n terms is known as the partial sum of the given infinite series. If S_n is given a limit as n moves towards infinity that limit is known as the sum to infinity of the series and the result is termed as the sum of infinity of series. 

Key Takeaways: Infinite series formula, sum of infinite geometric series, geometric progression, series infinity

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Infinite Series

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The arithmetic series is a sequence in which the difference between each consecutive term remains constant throughout, whereas the geometric series is a sequence in which the ratio of consecutive terms to the preceding term remains constant throughout.

A sequence a₁ , a₂ , a₃ ,…, a_n ,… is called arithmetic sequence or arithmetic progression if a_n + 1 = a_n + d, n ∈ N, where a₁ is known as the first term and the constant term d is referred to as the common difference of the A.P or Arithmetic Progression. 

Some properties of Arithmetic Progression are:

1. When each term of an A.P. is added up by a constant, the resulting sequence is also an A.P.
2. If you subtract a constant from each term of an A.P. The resulting sequence is also an A.P.
3. If an Arithmetic Progression's terms are multiplied by a constant, the resulting sequence is also an A.P.
4. If an Arithmetic Progression's terms are divided by a non-zero constant, the resulting sequence is also an A.P.

If the series has infinite terms, it is referred to as an infinite series, and the sum of the first n terms, S_n, is referred to as a partial sum of the given infinite series. If the partial sum, i.e. the sum of the first n terms, Sn, is given a limit as n approaches infinity, the limit is referred to as the sum to infinity of the series, and the result is referred to as the sum of the infinite series.

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Sum of Infinite Series Formula

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To find the sum of a series that extends to infinity, you can use the sum of the infinite geometric series formula. This is also referred to as the sum of infinite Geometric Progression. 

While calculating the sum of a GP, we discover that the sum converges to a value, despite the fact that the series has infinite terms. If the infinite series formula -1<r<1, could be given as, S\(\infty\)= \(\frac{a_1}{1-r}\) 

Here, 

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

If d > 0, the sum of an infinite arithmetic sequence is \(\infty\)

If d > 0- \(\infty\), d > 0, the sum of an infinite arithmetic sequence is \(\infty\)

Relationship between A.M. and G.M.

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Let A and G be A.M. and G.M. of two given positive real numbers a and b, respectively. 

Then A=\(\frac{a + b}{2}\)  and G = \(\sqrt{ab}\)

Thus, we have A – G =\(\frac{a + b}{2}\) –  \(\sqrt{ab}\) = \(\frac{a + b- 2\sqrt{ab}}{2}\)

= \(\frac{(\sqrt{a} - \sqrt{b})^2}{2}\)\( \geq\) 0

Therefore, we obtain the relationship A ≥ G.

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Things to Remember

• For an arithmetic series, the sum of infinite is undefined because the sum of terms leads to .
• If r is greater than one, the sum does not exist because it does not converge.
• If |r|< 1, the sum of a geometric series can be calculated to infinity.
• a= first term of the series, and r= a common ratio between two consecutive terms
• The formula for the sum of infinite series is, 

S\(\infty\)= \(\frac{a}{1-r}\) or \(\lim\limits_{n} \to \)\(\infty\)S_n = S\(\infty\)= \(\frac{a}{1-r}\)