(i) 1+1
2+1
22+1
23+1
24+··· (ii) 1−1
3+1
32−1
33+···(iii) 1+ 3 + 5 + 7 +··· (iv) 1+ 1 +1
2!+1
3!+1
4!+···(v) 1+1
2+1
3+1
4+···16.Obtain a formula for thenth partial sumSnfor the following series and hence in-
vestigate their convergence (or otherwise).
(i) 4+4
3+4
32+4
33+··· (ii) − 6 − 2 + 2 + 6 + 10 + 14 +···(iii) 1+ 2 + 22 + 23 +··· (iv) 1+x
2+(x2) 2
+(x2) 3
+···17.State, without proof, which of the following series are convergent/divergent. Write
down thenth term of each series.
(i) − 1 + 1 − 1 + 1 − 1 +··· (ii) 1. 01 +( 1. 01 )^2 +( 1. 01 )^3 +···(iii) (. 99 )^2 +(. 99 )^3 +(. 99 )^4 +··· (iv)1
50+1
51+1
52+1
53+···(v) 10^6 +105
2+104
3+103
4+··· (vi)1
40−1
50+1
60−1
70+···(vii)3
4+ 22(
3
4) 2
+ 32(
3
4) 3
+ 42(
3
4) 4
+···(viii)1
2+2
3+3
4+4
5+··· (ix) 1+(. 2 )+(. 2 )^2 +(. 2 )^3 +···18.Find thenth term of the following series and test for convergence.
(i) 1+ 1 + 1 + 1 +··· (ii) 1− 2 + 3 − 4 + 5 +···(iii) 1+ 2 (. 9 )+ 3 (. 9 )^2 + 4 (. 9 )^3 +··· (iv) 1−1
√
2+1
√
3−···(v) 1^3 + 23 (. 95 )+ 33 (. 95 )^2 +··· (vi)1
2+3
4+5
6+···(vii)1
2−3
4+5
6−7
8+··· (viii) 1·1
2+1
2·3
4+1
3·5
6+1
4·7
8+···19.Find the range of values forxfor which the following series are convergent.
(i) x−x^2
2+x^3
3−x^4
4+··· (ii) 2x+( 2 x)^2 +( 2 x)^3 +( 2 x)^4 +···(iii) 1+x+x^2
2!+x^3
3!+··· (iv) 1^5 + 25 x+ 35 x^2 + 45 x^3 + 55 x^4 +···(v) 1^2 + 22 x+ 32 x^2 + 42 x^3 +···20.For the binomial series( 1 +x)mshow that
∣
∣
∣
∣un+ 1
un∣
∣
∣
∣=∣
∣
∣
∣m−n+ 1
n∣
∣
∣
∣|x|and deduce that
the series converges if|x|<1 and diverges if|x|>1(mnot a positive integer).