1. Write the following in expanded for then find the sum
a) ^7(Sigma sign) k=0 (k=1)(k+2)
b) ^7(Sigma sign) k=4 3k
Note: The sigma sign does not show up when pasted here. The 7 is on top of the sigma sign and the k= 0 or k=2, respectively, lies below the sigma sign.

2. Express the following series using sigma notation:
a) (1 X 0)/2 + (2 X 1)/3 + (3 X 2)/4 + ........ (10 X 9)/11
b) (1 X 2)/2 + (2 X 3)/4 + (3 X 4)/6 + .......... (8 X 9)/16

3. Use the Binomial Theorem to expand the following equation:
a) (2p - 3q)^4
b) (2p + 1)^4

4. Find the fifth term in the expansion of (4x + 2y)^7

5. Find the sum of the following series, or state that the sum does not exist
a) 12 + 8 + 16/3 + .......
b) 1/12 + 1/2 + 3 + .........

6. Find the limit of the following or state that the limit does not exist
a) (2n - 1)/n^3
b) ......as n approches infinity of (2n^2)/(n^2 - 1)

Solution Preview

1. ∑_(k=0)^7▒(k+1)(k+2)
First, since this is summation from k=0 to k=7,
We could transform the expression into the following:
∑_(k=0)^7▒(k+1)(k+2) =(0+1)*(0+2)+(1+1)*(1+2)+(2+1)*(2+2)+(3+1)*(3+2)+(4+1)*(4+2)+(5+1)*(5+2)+(6+1)*(6+2)+(7+1)*(7+2)=1*2+2*3+3*4+4*5+5*6+6*7+7*8+8*9=240

2. ∑_(k=4)^7▒〖3k=3*4+3*5+3*6+3*7=66〗

3. a)(1*0)/2+(2*1)/3+(3*2)/4+⋯+(10*9)/11
First, in the first value, for the two numbers in the numerators, they are 1 and 2 less than denominator respectively.
Therefore, if k starts from 0, the first number a0=(0+1)*0/(0+2) or (k+1)k/(k+2) in general. ...

... Q(n) where P and Q are polynomial functions. ... Forty Problems involving Sequences, Series, Convergence, Divergence and ... listed in number 1 are geometric sequence. ...

...functions (linear, quadratic, rational, or exponential) is related to the geometric sequence? ... find at least two real-life examples of a sequences or series. ...

... Determine whether each of the following sequences are arithmetic ... the first fifty terms of the arithmetic sequence. ... Find the sum of the infinite geometric series. ...

... test to determine if the following sequences are monotonic ... Thus the sequence is monotonic decreasing on (1 ... term test to stae if the series converges, diverges ...

... continuous functions between open sets are 1-1 functions. ... Given that Then we define the series ∑ n with ... 1 is non-decreasing, bounded sequence => the sequence...

... Fundamental mathematics sequences are examined in the solution ... 1.15) is the famous Taylor series expansion of ... This sequence converges to the irrational number e ...

... Number sequences and series occur throughout the study of ... value of invested money can be computed with sequences. A certain sequence, 0.4, 0.7, 1, 1.6, 2.8, 5.2 ...

... particular solution in terms of familiar elementary functions. ... Determine whether or not the sequence converges and ... Use a geometric series to compute the total ...

... 2.Discuss how the different FUNCTIONS of DNA and PROTEIN ... is actually made up of a series of amino ... Additionally, the amino acid sequence of the protein will ...