MATH 1572H Spring 2013 Worksheet 4 Topic: limit of a sequence; basic properties of series. Two useful facts • For any sequence {an }, which goes to infinity, lim 1 + • For any sequence {bn }, which converges to zero, lim 1 an sin bn bn an = e. = 1. 1. For each of the given sequences, find its limit. 1 3n a) lim 1 + 2n b) lim n+1 n+3 n+2 sin 3 c) lim sin n2 n sin d) lim √ 1 n 1 n +1−1 Series Definition. Let {an } be a sequence. Then the expression of the form a1 + a2 + · · · + an + . . . is called a series. We attach a numerical value (”the sum“) to a series ∞ P an or, simply, n=1 ∞ P an by forming a the sequence of n=1 partial sums s 1 = a1 , s2 = a1 + a2 , s3 = a1 + a2 + a3 , sn = a1 + a2 + · · · + an , . . . and then finding its limit s = lim sn . It may happen that the limit does not exist. In such a case, we just say that the given series diverges. 1. For each of the given series, find a formula for its n-th partial sum sn and then find the sum of the series (if it exists). a) 1 + 0.1 + 0.01 + 0.001 + . . . b) 4 − 2 + 1 − 21 + 14 − 18 + . . . c) ∞ P n=1 3n +2n 5n d) x + 2x2 + 3x3 + 4x4 + . . . e) 1 2·4 + 1 3·5 + 1 4·6 + ··· + 1 n(n+2) + ... 2