Calculus 2

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FLORIDA INTERNATIONAL UNIVERSITY

MAC 2312 Online

Dr. George Kafkoulis, [email protected], DM410B tel: 3053482849

2. Assignment 2v02: On Properties of the summation notation

The objective behind the following three Problems is to make you use the properties of sums

and learn to break sums apart using simple algebra. I also want you to become fearless to symbols

and the use of mathematical abstraction

Problem 2.1. n2X

k=1

3k2 + 42k+1 + 54k�1

7�k+4 .

Problem 2.2. n3X

k=11

⇡2k + k2 + k

⇡2k+1k(k + 1) .

Problem 2.3. nX

k=1

3�k + 52k

7k .

Problem 2.4. nX

k=1

3 p k + 1 � 3

p k

3 p k2 + k

.

The objective behind this Problem is to make you use recognize hidden telescoping sums....

Problem 2.5. Compute the following sum (in terms of n, x):

n+20X

k=11

1

k(k + 1)(k + 2)(k + 3) .

The objective behind this Problem is to make you use recognize hidden telescoping sums....

Problem 2.6. Compute the following sum:

n2X

k=n

3[sin(e ek+12)] � 3[sin(e

ek+13)].

The objective behind this Problem is to make you use all the summation properties with no fear.

Sometimes things look more complicate than they really are....

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Problem 2.7. Compute each of the following sums:

240X

j=11

j + 32�7j + 55j�1

4�3j�2 ,

1999X

j=1

cos(5j+31) � cos(5j+30)

The objective behind this Problem is to make you use all the summation properties.

Problem 2.8. Calculate the general formula for the following sums:

(i) nX

k=1

k4 (ii) nX

k=1

k5.

The objective behind this Problem is to make you use special sums.

Problem 2.9. Calculate the general formula for the following sums:

(i) nX

k=1

((k + 5)2 (ii) nX

k=m

(k + 3)3 (iii) n2X

k=m2

(k2 + 3k + 1)2.

The objective behind this Problem is to make you recognize special sums and the use of

translation invariance.

Problem 2.10. Calculate the general formula for the following sums:

(i) nX

k=m

(�1)k 2 x2k (ii)

nX

k=m

(�1)k 2+1x3k+2.

This is a tricky, but simple problem.... The objective behind this Problem is to make you recognize Geometric Sums.

Problem 2.11. Calculate the general formula (in terms of m, n) for the following sums:

nX

k=m

((k � 3)2 � 1).(2.1)

nX

k=4

32k+1 � 27k�2 � 5�6k

511k�111 .(2.2)

The objective behind this Problem is to make your learn the use of Linearity and Geometric

Sums.

Problem 2.12. (The Glossary: Definitions and Theorems) State precisely the following definitions and theorems.

(1) The Linearity Property Theorem. (2) The Telescoping Property Theorem. (3) The Geometric Property Theorem. (4)

Pm k=1 k = ...

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(5) Pm

k=1 k 2 = ...

(6) Pm

k=1 k 3 = ...

The objective behind this Problem is to make you learn the glossary of mathematical terms

behind this module and assignment. When you learn the precise definitions and theorems, then

we can communicate e↵ectively and you will be able to understand faster and easier the concepts

I teach you.