Infinite series first video, bounded sequence, concept of convergence divergence and oscillatory series
Prove that the series a+ar+ar²+..... Which is a series in GP is convergent only when common ratio r<1
If k is a finite number and ∑ uⁿ Converse to a sum s then ∑ kuⁿ also Converse and its sum is ks also if the first series diverse or oscilates so does the second unless k=0
if the series u¹+u²+u³.....and v¹+v²+v³+... converge and their sums are s and t respectively then
- (u¹+v¹)+(u²+v²)+(u³+v³)+........ Converse and its sum is s+t.
- (u¹-v¹)+(u²-v²)+u³-v³)+........ Diverges and its sum is s-t.
If a finite number of terms from a given series is omitted (or added) the convergence or divergence of the series is not affected.
series of positive terms.
(i) series of positive terms u¹+u²+u³+.....uⁿ+..... Is convergent if a number k can be found such that sⁿ=u¹+u²+u³+........uⁿ < k for all n.
(ii) if no such number k can be found then the series is divergent.
if each term of an infinite series of positive terms is greater than a finite quantity however a small then the series is divergent.
if each term of an infinite series of positive terms is greater than a finite quantity however a small then the series is divergent.