h2 summation

lin2 SG

hello

1.can i know how do u change Σ (upperlimit: N, lowerlimit: n=1) [(2n+1)/(n^2(n+1)^2)] to Σ (upperlimit: N, lowerlimit: n=2) [(2n-1)/(n^2(n-1)^2)] ?

2. Show that if Ur = 1/(2r+1), then U(r-1) +Ur= 4r/(4r^2-1). Using the results or otherwise, find Σ (upperlimit: 2n , lowerlimit: r=2) [(-1)^r (r/(4r^2-1))] i do not know how to do the second part.

3. show that (nC4)+(nC5)=((n+1)C5) for all n≥5. Hence, or otherwise, determine the largest interger value of N such that Σ(upperlimit:N lowerlimit:n=5) (nC4) < (1000 C 5)

wee_ws SG

Hi,

Please give details of Q1. The context of the problem is absent.

In Q3, please highlight the specific difficulty you are facing.

For Q2, which is about an alternating series, we have

Σ (upperlimit: 2n , lowerlimit: r=2) [(-1)^r (r/(4r^2-1))]

= 1/4 . Σ (upperlimit: 2n , lowerlimit: r=2) [(-1)^r {U_{r-1} + U_r}]

= 1/4 [ (U_1 + U_2)

- (U_2 + U_3)

+ (U_3 + U_4)

- ...

+ (U_{2n-1} + U_{2n} ]

= ?

You should be able to complete the rest of the steps after systematic cancellation of terms.

Thanks.

Cheers,
Wen Shih

lin2 SG

For qn 3, I cannot get the second part. (determine the largest interger value of N such that Σ(upperlimit:N lowerlimit:n=5) (nC4) < (1000 C 5))

For the first one,

A sequence, U1,U2,U3,… is such that Un = 1/(n^2) and Un+1= Un- (2n+1)/(n^2(n+1)^2), for all n≥1.

i) Find Σ(upper limit: N lower limit:n=1) (2n+1)/(n^2(n+1)^2)

ii) Give a reason why the series in parti is convergent and state the sum to infinity

iii) Use your answer to part (i) to find Σ (upperlimit: N, lowerlimit: n=2) [(2n-1)/(n^2(n-1)^2)]

I do not know how to do part iii only.

wee_ws SG

Hi,

For Q1(iii), expand the sum

Σ (upperlimit: N , lowerlimit: n=2) (2n-1)/(n^2 (n-1)^2)

= 3/(2^2.1^2) + 5/(3^2.3^2) + ... + (2N-1)/(N^2 (N-1)^2).

Now express this expansion in sigma notation using the general term in (i), i.e.,

(2n+1)/(n^2 (n+1)^2).

You should be able to determine the lower and upper limits by looking at the terms

3/(2^2.1^2) and (2N-1)/(N^2 (N-1)^2).

This practical strategy always works :)

For Q3, I believe you would have arrived at the inequality (please show working next time)

(N+1)C5 - 1 < 1000C5.

It is clear what the largest possible N is, ya?

Thanks.

P.S. I made a mistake which has been corrected :P

Cheers,
Wen Shih