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*