Engineering Math

1. SERIES
•Definition & Types of Sequence
•A sequence is a set of quantities stated in a
definite order and each term formed
according to a fixed pattern;
•1, 3, 5, 7,………
•2, 6, 18, 54,……

2. SERIES
•Definition & Types of Sequence
•A finite sequence contains only a finite
number of terms;
•The page numbers of a book.
•An infinite sequence is unending;
•All the natural numbers.

3. SERIES
•Definition & Types of Sequence
•A series is formed by the sum of the terms
of a sequence;
•1 + 3 + 5 + 7 +……..
•𝑢𝑛 = 𝑛𝑡ℎ
term
•𝑆𝑛 =sum of the first ‘n’ terms

4. SERIES
•Definition & Types of Sequence
•Arithmetic Progression
•2 + 5 + 8 + 11 + 14 +……….
•𝑎 + 𝑎 + 𝑑 + 𝑎 + 2𝑑 + 𝑎 + 3𝑑 +……..
• 𝑎 =first term 𝑑 =common difference
•𝑛𝑡ℎ
𝑡𝑒𝑟𝑚 = 𝑎 + 𝑛 − 1 𝑑
•𝑆𝑛 =
𝑛
2
(2𝑎 + 𝑛 − 1 𝑑)

5. SERIES
•Definition & Types of Sequence
(1) Find the sum of the first 20 terms of the
series: 10 + 6 + 2 − 2 − 6……
(2) If the seventh term of an AP is 22 and the
twelfth term is 37, find the series.
(3) If the sixth term of an AP is -5 and the tenth
term is -21, find the sum of the first 30 terms.

6. SERIES
•Definition & Types of Sequence
•Geometric Progression
•1 + 3 + 9 + 27 + 81 +……….
•𝑎 + 𝑎𝑟 + 𝑎𝑟2
+ 𝑎𝑟3
+…..
•𝑎 =first term r =common ratio
•𝑛𝑡ℎ
𝑡𝑒𝑟𝑚 = 𝑎𝑟𝑛−1
•𝑆𝑛 =
𝑎(1−𝑟𝑛)
1−𝑟

7. SERIES
•Definition & Types of Sequence
•Geometric Progression
(1) For the series 8 + 4 + 2 + 1 +
1
2
+……, find
the sum of the first eight terms.
(2) If the fifth term of a GP is 162 and the eighth
term is 4374, find the series.

8. SERIES
•Definition & Types of Sequence
•Geometric Progression
(3) For the series in question (2) above, find:
(a) the tenth term
(b) the sum of the first ten terms.
(4) 𝑥 + 1 , 𝑥 + 3 𝑎𝑛𝑑 (𝑥 + 8) are the first
three terms of a GP. Find 𝑎 𝑥 𝑏 𝑟

9. SERIES
•Definition & Types of Sequence
•The sum of the first ‘n’ terms of a GP is;
𝑆𝑛 = 𝑎 + 𝑎𝑟 + 𝑎𝑟2
+ 𝑎𝑟3
+ ⋯ + 𝑎𝑟𝑛−2
+ 𝑎𝑟𝑛−1
•Multiply through by 𝑟
𝑟𝑆𝑛 = 𝑎𝑟 + 𝑎𝑟2
+ 𝑎𝑟3
+ ⋯ + 𝑎𝑟𝑛−1
+ 𝑎𝑟𝑛
Subtract the two equations:
𝑆𝑛 − 𝑟𝑆𝑛 = 𝑎 − 𝑎𝑟𝑛

10. SERIES
•Definition & Types of Sequence
•𝑆𝑛(1 − 𝑟) = 𝑎 1 − 𝑟𝑛
•𝑆𝑛 =
𝑎(1−𝑟𝑛)
1−𝑟
for 𝑟 < 1; or
•𝑆𝑛 =
𝑎(𝑟𝑛−1)
𝑟−1
for 𝑟 > 1

11. SERIES
•Definition & Types of Sequence
•If 𝑟 < 1, then 𝑟𝑛
→ 0
•𝑛 →
𝑆𝑛
∞ =
𝑎(1−0)
1−𝑟
•𝑆∞ =
𝑎
1−𝑟

12. SERIES
•Limits of Sequences & its Properties
•Limit of a Function
•The limit of the function 𝑓(𝑥) as 𝑥
approaches the value 𝑎, written as
lim
𝑥→𝑎
𝑓 𝑥 = 𝑓(𝑎)is obtained by replacing
𝑥 by 𝑎 in the function. It follows that
lim
𝑥→𝑎
𝑓 𝑥 = 𝑓(𝑎)

13. SERIES
•Limits of Polynomial Functions
•𝑓 𝑥 = 𝑎0𝑥𝑛 + 𝑎1𝑥𝑛−1 + 𝑎2𝑥𝑛−2 +
⋯ + 𝑎𝑛−1𝑥 + 𝑎𝑛
•(1) lim
𝑥→3
𝑥 + 4
•(2) lim
𝑥→4
4𝑥2 − 12𝑥 + 4

14. SERIES
•Limits of Polynomial Functions
(3) lim
𝑥→−2
2𝑥3 + 3𝑥2 + 5𝑥 + 6
(4) lim
𝑥→4
𝑥2 − 3𝑥 + 5

15. SERIES
•Limits of Rational Functions
•A function 𝑓(𝑥) is said to be a
rational function if it can be
expressed as the quotient of two
polynomials in 𝑥; that is:

16. SERIES
•Limits of Rational Functions
• 𝑓 𝑥 =
𝑔(𝑥)
ℎ(𝑥)
for ℎ(𝑥) ≠ 0
•The limit as 𝑥 approaches a is given by:
lim
𝑥→𝑎
𝑓 𝑥 = 𝑓 𝑎 =
𝑔(𝑎)
ℎ(𝑎)

17. SERIES
•Limits of Rational Functions
• If both 𝑔 𝑎 𝑎𝑛𝑑 ℎ(𝑎) are both
zero or infinity, then we manipulate
to reduce it to its simplest form.

18. SERIES
•Limits of Rational Functions
(1) lim
𝑛→∞
4
𝑛
=
4
∞
= 0
(2) lim
𝑥→0
3
𝑥
=
3
0
= ∞
(3) lim
𝑛→0
𝑛
7
=
0
7
= 0

19. SERIES
•Exercise
(1)lim
x→2
x2−4
x−2
(2) lim
x→3
x−3
x2−5x+6
(3) lim
𝑥→0
2𝑥3−3𝑥2+4𝑥
5𝑥3+2𝑥2−3𝑥
(4) lim
𝑥→0
8𝑥+9𝑥2
𝑥(𝑥2−9)

20. SERIES
•Exercise
(5) lim
x→0
(x−2)2−(2x+1)(x+4)
x(2x+3)
(6) lim
x→16
16−x
4− 𝑥
(7) lim
𝑥→0
3𝑥−4 2𝑥+3 −(5𝑥−2)(𝑥+6)
𝑥(𝑥2−9)
(8) lim
𝑥→3
9−𝑥2
4− 𝑥2+7

21. SERIES
•Infinity
(1) lim
x→∞
2𝑥2−3x
3𝑥2−2
(2) lim
x→∞
4𝑥3−5𝑥2
2𝑥3+𝑥
(3) lim
𝑥→∞
5𝑥
𝑥+2
(4) lim
𝑥→∞
𝑥3−1
𝑥−1

22. SERIES
•Recall the following derivatives
(1)
𝑑
𝑑𝑥
sin 𝑥 = cos 𝑥
(2)
𝑑
𝑑𝑥
cos 𝑥 = −sin 𝑥
(3)
𝑑
𝑑𝑥
tan 𝑥 = 𝑠𝑒𝑐2 𝑥

23. SERIES
•Recall the following derivatives
(4)
𝑑
𝑑𝑥
𝑒𝑎𝑥 = 𝑎𝑒𝑎𝑥
(5)
𝑑
𝑑𝑥
ln 𝑎𝑥 =
𝑎
𝑎𝑥

24. SERIES
•Recall the following trig identities
(1)
1
sin 𝑥
= 𝑐𝑜𝑠𝑒𝑐 𝑥
(2)
1
cos 𝑥
= 𝑠𝑒𝑐 𝑥
(3)
1
tan 𝑥
= 𝑐𝑜𝑡 𝑥

25. SERIES
•Trigonometric Functions
(1) lim
x→0
sin 𝑥 (2) lim
x→0
cos 𝑥
(3) lim
𝑥→0
𝑠𝑖𝑛𝑥
𝑥
(4) lim
𝑥→0
sin 3𝑥
𝑥

26. SERIES
•Trigonometric Functions
(5) lim
x→0
tan 𝑥
𝑥
(6) lim
x→0
cot 𝑥
cot 2𝑥
(7) lim
𝑥→0
𝑥
tan 5𝑥
(8) lim
𝑥→0
𝑥 𝑐𝑜𝑠𝑒𝑐 𝑥

27. SERIES
•Trigonometric Functions
(9) lim
x→0
sin 3𝑥
sin 8𝑥
(10) lim
x→0
tan 3𝑥
𝑥

28. SERIES
•Indeterminate Forms
(1) lim
x→2
𝑥2+5𝑥−14
𝑥2−5𝑥+6
(2) lim
x→0
tan 𝑥 −𝑥
𝑥3
(3) lim
x→0
𝑠𝑖𝑛2 𝑥
𝑥2

29. SERIES
•L’Hopital’s Rule
•Suppose we have to find the limiting value of a
function
𝑓(𝑥)
𝑔(𝑥)
at 𝑥 = 𝑎, when direct substitution
of 𝑥 = 𝑎 gives the indeterminate form
0
0
, we
find the ratio of the derivatives of numerator
and denominator at 𝑥 = 𝑎.

30. SERIES
•L’Hopital’s Rule
• lim
x→a
𝑓(𝑥)
𝑔(𝑥)
=
𝑓′(𝑎)
𝑔′(𝑎)
= lim
x→a
𝑓′(𝑥)
𝑔′(𝑥)
•L’Hopital’s rule applies only when the indeterminate
form arises. If the limiting value can be found by
direct substitution, the rule will not work.

31. SERIES
•L’Hopital’s Rule
•Consider; lim
x→2
𝑥2+4𝑥−3
5−2𝑥
•Direct substitution = 9
•L’Hopital’s rule = -4

32. SERIES
•L’Hopital’s Rule
•Before using L’Hopital’s rule, you must satisfy
yourself that direct substitution gives the
indeterminate form
0
0
. If it does, you may use the
rule, but not otherwise.
•The rule can be used twice, thrice,etc..

33. SERIES
•L’Hopital’s Rule
(1) lim
x→1
𝑥3+𝑥2−𝑥−1
𝑥2+2𝑥−3
(2) lim
x→0
𝑥−sin 𝑥
𝑥2
(3) lim
x→1
𝑥3−2𝑥2+4𝑥−3
4𝑥2−5𝑥+1

34. SERIES
•L’Hopital’s Rule
(4) lim
x→0
tan 𝑥−𝑥
sin 𝑥−𝑥
(5) lim
x→0
𝑥 cos 𝑥−sin 𝑥
𝑥3

35. SERIES
•Finding a formular for 𝑢𝑛, and its
limiting value:
(1) 1,
1
2
,
1
4
,
1
8
,
1
16
, … …
(2) 1, -1, 1, -1, ……..
(3) 1,
1
2
,
1
3
,
1
4
,
1
5
, … …

36. SERIES
•Finding a formular for 𝑢𝑛, and its
limiting value:
(4) 1, 0, 1, 0, 1, 0, ……..
(5) 0.9, 0.99, 0.999, 0.9999, …….
(6) 1, 2,
3
3,
4
4, … …