Maths Home Assignment 3

M102

ASSIGNMENT IV

A. Mookerjee

1. If q and q 0 are rational numbers and a is a positive real number, show that (a)ax ·ay = ax+y and (b) (ax )y = ax·y Let (xn ) be a sequence of rational numbers such that limn→∞ xn = 0, show that for a(> 0) ∈ < lim axn = 1 n→∞

2. Show that the sequence (xn ) with x ∈ < is convergent if and only if −1 < x ≤ 1 3. Show that lim n1/n = 1

n→∞

4. If (an ) with a ∈ < is a sequence. Show that lim

n→∞

an+1 = ` |`| < 1 ⇒ limn→∞ an = 0 an

Hence show that lim(xn /n!) = 0 for any x ∈ <. 5. If the sequences (an ) and (bn ) tend to 0 and if the latter is a strictly monotonic decreasing sequence of positive real numbers, then lim n→∞

an − an+1 an = n→∞ lim bn bn − bn+1

Provided the right hand side exists. 6. (an ) iis any sequence and (bn ) is a strictly monotonic increasing sequence of real numbers and bn → ∞, then lim

n→∞

an an − an+1 = lim n→∞ b − b bn n n+1

Provided the right hand side exists. 7. If lim an = ` then lim

n→∞

a1 + a 2 + . . . + a n = ` n 1

8. If xn > 0 and xn+1 = (xn + xn−1 )/2, then show that the sequence (x2n+1 ) is a decreasing and (x2n ) is an ncreasing sequence and they both coverge to the same limit (x1 + 2x2 )/3 √ 9. If k, a1 > 0 and an+1 = k + an show that the sequence (an ) increases or decreases acording to whether a1 is less than or greater than the positive root of x2 −x−k = 0. In either case this root is its limit.

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