Question 1001038
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Does the infinite series diverge or converge? If it converges what is the sum?
10+2+2/5+2/25+....
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It is the sum a geometric progression with the first term a = 10 and the common ratio r = {{{1/5}}}.


This infinite series converges for any geometric progression with the <U>common ratio less than 1 in the modulus</U>.


The sum of an infinite geometric progression with the first term a and the raio r, |r| < 1, is 


S = {{{a/(1-r)}}}.


In your case S = {{{10/(1 - (1/5))}}} = {{{10/((4/5))}}} = {{{(10*5)/4}}} = 12.5.


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what is 
a_n= 2a_n-1 -1 where a_1 = 2? 
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{{{a[1]}}} = 2

{{{a[2]}}} = {{{2*a[1] - 1}}} = 2*2 - 1 = 3,

{{{a[3]}}} = {{{2*a[2] - 1}}} = 2*3 - 1 = 5,

{{{a[4]}}} = {{{2*a[3] - 1}}} = 2*5 - 1 = 9,

{{{a[5]}}} = {{{2*a[4] - 1}}} = 2*9 - 1 = 17,

and so on . . . 


You can easily calculate it yourself.