Question Number 672 by 123456 last updated on 21/Feb/15
![if a_n ,b_n ,c_n are real sequence with a_n >0,b_n >0,c_n >0 and a_n ^n <b_n <c_n ^(1/n) if Σ_(n=0) ^(+∞) a_n converge and Σ_(n=0) ^(+∞) c_n converge did Σ_(n=0) ^(+∞) b_n converge?](https://www.tinkutara.com/question/Q672.png)
$${if}\:{a}_{{n}} ,{b}_{{n}} ,{c}_{{n}} \:{are}\:{real}\:{sequence}\:{with} \\ $$$${a}_{{n}} >\mathrm{0},{b}_{{n}} >\mathrm{0},{c}_{{n}} >\mathrm{0} \\ $$$${and} \\ $$$${a}_{{n}} ^{{n}} <{b}_{{n}} <{c}_{{n}} ^{\mathrm{1}/{n}} \\ $$$${if}\:\underset{{n}=\mathrm{0}} {\overset{+\infty} {\sum}}{a}_{{n}} \:{converge}\:{and}\:\underset{{n}=\mathrm{0}} {\overset{+\infty} {\sum}}{c}_{{n}} \:{converge} \\ $$$${did}\:\underset{{n}=\mathrm{0}} {\overset{+\infty} {\sum}}{b}_{{n}} \:{converge}? \\ $$
Commented by prakash jain last updated on 22/Feb/15
![For Σ_(n=1) ^∞ a_n ^n lim_(n→∞) (a_n ^n )^(1/n) =lim_(n→∞) a_n =0 so this converges. For Σ_(n=1) ^∞ c_n ^(1/n) assume c_n =(1/n^2 ) c_n ^(1/n) does not converge. Σ_(n=1) ^∞ b_n cannot conclude as convergent or divergnt.](https://www.tinkutara.com/question/Q675.png)
$$\mathrm{For}\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}{a}_{{n}} ^{{n}} \\ $$$$\underset{\mathrm{n}\rightarrow\infty} {\mathrm{lim}}\sqrt[{{n}}]{{a}_{{n}} ^{{n}} }=\underset{{n}\rightarrow\infty} {\mathrm{lim}}{a}_{{n}} =\mathrm{0}\: \\ $$$$\mathrm{so}\:\mathrm{this}\:\mathrm{converges}. \\ $$$$\mathrm{For}\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}{c}_{{n}} ^{\mathrm{1}/{n}} \\ $$$$\mathrm{assume}\:{c}_{{n}} =\frac{\mathrm{1}}{{n}^{\mathrm{2}} } \\ $$$${c}_{{n}} ^{\mathrm{1}/{n}} \:\mathrm{does}\:\mathrm{not}\:\mathrm{converge}. \\ $$$$\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}{b}_{{n}} \:\mathrm{cannot}\:\mathrm{conclude}\:\mathrm{as}\:\mathrm{convergent} \\ $$$$\mathrm{or}\:\mathrm{divergnt}. \\ $$