Question

a) Use the Ratio Test to show that the series ( sum_{n=0}^{infty} frac{1}{n !} ) converges. b) Use the Ratio Test to show that for any value of ( x ) the series ( sum_{n=0}^{infty} frac{x^{n}}{n !} ) will converge absolutely. (Hint: In applying the ratio test assume that ( x eq 0 ) is fixed and that ( a_{n}=frac{|x|^{n}}{n !} ).) c) Use the Ratio Test to show that the series ( sum_{n=1}^{infty} frac{x^{n}}{n} ) will converge absolutely if ( |x|<1 ) and it diverges when ( |x|>1 ). d) What happens to the convergence of the series ( sum_{n=1}^{infty} frac{x^{n}}{n} ) when ( x=1 ) and when ( x=-1 ). (Substitute ( x= pm 1 ) in for ( x ) and then determine the convergence or divergence of the individual numerical series.)

          a) Use the Ratio Test to show that the series ( sum_{n=0}^{infty} frac{1}{n !} ) converges.
b) Use the Ratio Test to show that for any value of ( x ) the series ( sum_{n=0}^{infty} frac{x^{n}}{n !} ) will converge absolutely. (Hint: In applying the ratio test assume that ( x 
eq 0 ) is fixed and that ( a_{n}=frac{|x|^{n}}{n !} ).)
c) Use the Ratio Test to show that the series ( sum_{n=1}^{infty} frac{x^{n}}{n} ) will converge absolutely if ( |x|<1 ) and it diverges when ( |x|>1 ).
d) What happens to the convergence of the series ( sum_{n=1}^{infty} frac{x^{n}}{n} ) when ( x=1 ) and when ( x=-1 ). (Substitute ( x= pm 1 ) in for ( x ) and then determine the convergence or divergence of the individual numerical series.)

$a) Use the Ratio Test to show that the series ( sumn=0^infty frac1n ! ) converges. b) Use the Ratio Test to show that for any value of ( x ) the series ( sumn=0^infty fracx^nn ! ) will converge absolutely. (Hint: In applying the ratio test assume that ( x eq 0 ) is fixed and that ( an=frac|x|^nn ! ).) c) Use the Ratio Test to show that the series ( sumn=1^infty fracx^nn ) will converge absolutely if ( |x|<1 ) and it diverges when ( |x|>1 ). d) What happens to the convergence of the series ( sumn=1^infty fracx^nn ) when ( x=1 ) and when ( x=-1 ). (Substitute ( x= pm 1 ) in for ( x ) and then determine the convergence or divergence of the individual numerical series.)$

Added by Jing W.

Question

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