Consider
Series :
Root
Test: Consider a infinite term series .
Let
, Then :
(i) if L<1 then series is convergent.
(ii) if L>1 then series is divergent
(iii) if L=1 then can not say anything means test fails.
Here ,
Now,
Hence, by Root Test this series is Convergent.
Consider
Series:
Ratio
Test: Consider a infinite term series
.
Let
, Then :
(i) if L<1 then series is convergent.
(ii) if L>1 then series is divergent
(iii) if L=1 then can not say anything means test fails.
Here,
So,
Now,
{ in first term 1/2 <1 so (1/2)^n will zero if n tends to
infinity. In second term denominator will become infinity if n
tends to infinity so whole term will be zero}
Hence, by Ratio Test Series is convergent.
Consider
Series:
Here,
Now,
Hence by Ratio Test , series is convergent.
Consider
Series:
Necessary Condition for convergence of a series is that
.
Here,
Now,
So, from above condition this series not convergent.
Final Result:
series1 | Convergent | Root Test |
series2 | Convergent | Ratio Test |
series3 | Convergent | Ratio Test |
series4 | Not Convergent | Necessary Condition Test |
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