Determine whether the Geometric series is Convergent or Divergent. 10 − 4 + 1.6 − 0.64 + ….

determine whether the geometric series is convergent or diverge

This question aims to find whether the given series falls in the category of convergent or divergent. The given series is:

\[ S = 10 – 4 + 1.6 – 0.64 . . . \]

In mathematics, a series is the sum of all the values in the sequence. We can get a series by adding infinitely many quantities one by one to the first-mentioned quantity. These types of series are also called infinite series. They are represented by $ a_i $. The addition of infinite quantities can be described by the expression:

\[ a_1 + a_2 +a_3 + . . . \]

\[ \sum_{i=1}^\infty \]

It is practically impossible to have the sum of infinite quantities. Instead of saying infinite quantities, we simply take finite sums of the $n$ starting terms of the series. This is also called the partial sum of the series.

\[ \sum_{i=1}^\infty  a_i=  \lim_{n\to\infty} \sum_{i=1}^n  a_i\]

Expert Answer

When the terms in the series fulfill the requirement of the above-mentioned limit then it means the series is convergent and we can take the sum of these series. but if the series is not summable then we will say that it is a divergent series.

We can take the geometric sum of the series by following formula:

\[ S_n = \frac { a_1 } { 1 – r } \]

Where $ a_1 $ is the first term of the series and $ r $ is the common ratio. To correctly find the common ratio, divide the second term by the first term of the series.

\[ r = \frac {a_2} {a_1} \]

First term is $ 10 $ and the second term is $ -4 $ in the given series. Hence,

\[ r = \frac { -4 } { 10 } \]

\[ r = \frac { -2 } { 5 } \]

By using values in the formula of geometric series:

\[ S_n = \frac { 10 } { 1 – (\frac{-2 } {5})} \]

\[ S_n = \frac { 50 } { 7 } \]

Numerical Solution

The sum of given series is $ \frac { 50 } { 7 } $ . The given series is summable which is why it is a convergent series. 

Example

A series is called convergent when its common ratio is less than $ 1 $

\[| r | < 1\]

\[ S = 10 – 3 + 1.6 – 0.64 . . . \]

The geometric series are written in the form of:

\[ S = a + ar + ar^2 + . . . \]

\[ \frac { a } { 1 – r } = a + ar + ar^2 + . . . \]

Where $ a $ is the first term of the series and $ r $ is the common ratio.

\[ r = \frac {a_2} {a_1} \]

\[r = \frac { -3 } { 10 }\]

\[r = – 0.3\]

\[r < 1\]

\[- 0.3 < 1\]

It means the given geometric series is convergent.

Images/Mathematical drawings are created in Geogebra

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