Lesson 8

## Infinite Series

This is a series with infinity number of terms.

NB. If one added infinity number of terms of an arithmetic series, the sum becomes a very large positive number which increases as more and more terms are added e.g. 2 + 4 + 6 + 8 + 10 +……..

Since the sum of terms increases towards a very large positive hence the sum to infinity does not exist. We call that series a Divergent series.

NB. An infinity geometric series converges under certain circumstances. Consider the following cases. CASE 2. Geometric series 𝑟 < 1 e.g. 𝑟 = 2 If 𝑎 = 1 ; ∴ 1 + 2 + 4 + 8 + 16 +…….𝑆1 = 1𝑆2 = 3𝑆3 = 7𝑆4 = 15As more and more terms are added the 𝑆𝑛 becomes bigger and bigger (large positive). Since the sum is not approaching a particular value (figure) then the series is said to diverge and its sum to infinity does not exist. ## Definition Of A Series

### If 𝑇1; 𝑇2; 𝑇3; … … … 𝑇𝑛 denotes a sequence of which the 𝑛𝑡ℎ term is 𝑇𝑛 , then the 𝑇1 + 𝑇2 + 𝑇3 … … … + 𝑇𝑛 is called a series.

A series is given by adding terms of a particular sequence.

Symbols

𝑆𝑛 Denotes the sum of the first in terms

𝑆10 means the sum of the first 10 terms in the sequence.

𝑆10 = 𝑇1 + 𝑇2 + 𝑇3 … … … + 𝑇10

## HTML Lesson

Full explanation on this topic and how to answer questions in an exam.

## Mechanical Energy (Lesson 7)

1. Prove (accepting results established in earlier grades):
• That a line drawn parallel to one side of a triangle divides the other two
sides proportionally (and the Midpoint Theorem as a special case of the converse of this theorem);
• That equiangular triangles are similar;
• That triangles with sides in proportion are similar; and the Pythagorean
Theorem by similar triangles

1. Revise the following including grade 10 concepts:
• The equation of a line through two given points;
• The equation of a line through one point and parallel or perpendicular to
a given line; and
• The inclination (θ) of a line, where 𝑚 = tan 𝜃 is the gradient of the line (0° ≤ 𝜃 ≤ 180°)