MA1511 Cheat Sheet V1 - Summary Engineering Calculus National University of Singapore

MA1511 Engineering Calculus compiled by Elston
� Infinite Series
�.� Partial Sum and Infinite Sums
Partial Sum
{S
n
} = a
1
+ a
2
+ a
3
+ … + a
n
=
P
n
k=1
a
k
.
If
P
1
k=1
a
k
converges, lim
n!1
S
n
= L. Else, S
n
diverges.
�.� Geometric Series
P
n
k=1
ar
k1
=
a(1r
n
)
1r
⇣
=
a
1r
if |r| < 1 and n !1
⌘
�.� Telescope Series
Series in the form of
P
1
k=1
(
U
k
 U
k1
)
converges.
�.� Useful Results for Limits
lim
n!1
a
1
n
=1,a, 0.
lim
n!1
n
1
n
=1
lim
n!1
r
n
=0, |r| < 1
lim
n!1
✓
1+
a
n
◆
n
= e
a
8a.
lim
n!1
n
a
a
n
=0,a>0
lim
n!1
lnn
n
a
=0
lim
n!1
a
n
n!
=0
lim
n!1
n!
n
n
=0
Order of Magnitude: n
n
>n! >a
n
>n
a
> ln n
�.� Convergence Tests
Test Conditions Conclusion Types of Series
n
th
term
divergence test
lim
n!1
a
n
, 0
lim
n!1
a
n
=0
diverges
No conclusion
1
X
i
a
n
,
p-series test
1
X
n
1
n
p
(p-series)
converges () p>1
p-series
Ratio Test*
lim
n!1
a
n+1
a
n
= c,c < 1
c>1
c =1
converges
diverges
no conclusion
Factorials [n!]
Polynomials [n
2
]
Exponentials [c
n
]
Root Test*
lim
k!1
a
k
1
k
= c,c < 1
c>1
c =1
converges
diverges
no conclusion
Exponentials [c
n
]
k
a
,a2 R
Alt. Series
Test
i. a
k
> 08k
ii. a
k
is decreasing
iii. lim
k!1
a
n
=0
converges
(diverges if iii.
false)
i.
1
X
k
(1)
k
a
k
ii.
1
X
k
(1)
k1
a
k
*Used to test for absolute convergence.
i.e. lim
k!1
|a
k
1
k
| = c
or lim
n!1
|
a
n+1
a
n
| = c
�.� Power Series
Power Series:
P
1
k=0
c
k
(
xa
)
k
,
a
is the center and
c
k
are the coefficients.
1. Find absolute convergence. (|L| < 1 converges)
2. Form
|x  a| <R
, where R is the radius. Then find interval of
convergence.
Special Cases:
– L =0: Series converges for all x. Radius = 1
– L = 1: Series diverges 8x except x = a. Radius = 0
– L<1: Find limits by solving for inequality.
Note: test endpoints to see whether
a
k
fulfils ratio/root test. Include
the value into the limits.
� Partial Derivatives
�.� Level Curves
The projection of the contour curve onto the
x  y
plane is a level
curve of
f
. Hence, a level curve consists of the set of points (
x,y
) for
which f (x,y) has a constant value.
�.� Normal Lines and Tangent Planes
Let a surface
z
=
f
(
x,y
) and a point
P
(
a,b,f
(
a,b
)) on the surface given.
Tangent Eqn : r ·
0
B
B
B
B
B
@
f
x
(a,b)
f
y
(a,b)
1
1
C
C
C
C
C
A
=
0
B
B
B
B
B
@
f
x
(a,b)
f
y
(a,b)
1
1
C
C
C
C
C
A
·
0
B
B
B
B
@
a
b
f (a,b)
1
C
C
C
C
A
Normal Eqn :
0
B
B
B
B
@
x
y
z
1
C
C
C
C
A
=
0
B
B
B
B
@
a
b
f (a,b)
1
C
C
C
C
A
+
0
B
B
B
B
B
@
f
x
(a,b)
f
y
(a,b)
1
1
C
C
C
C
C

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