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approach  each  other  without  limit  as  the  number  of  terms  is  increased;
and  find  at  what  place  of  decimals  they  differ  when  32  terms  are
taken.
In  the  first  series  a  =  1,  »' «  f ;

As  n  increases  without  limit,  (^)*  becomes  indefinitely  smaU,
hence  8^  approaches  5  without  limit.
In  the  second  series  a  »  f ,  r  «  | ;

As  n  increases  without  limit,  (|)"  becomes  indefinitely  small,
hence  8^  approaches  5  without  limit.
Since  the  sums  of  both  series  approach  the  same  limiting  value,
they  approach  each  other  without  limit.
When  n  =  32,
for  the  first  series  j     ,  gx,,  ^  53  x  1.9081

S.  =  5-5(.8)«  "*'        ^4,9

=  5-.004;  .8'«=.0008

for  the  second  series

5„  =  5  -  5  (.5)»«  log  (-Sp  =  \jm^  X  82

=  5 -.000000001,  =10.8

.*.    when  fi »  82,  the  two  sums  differ  at  the  8rd  decimal  place.
264.  Gtoneral  formula  t&r  sum  to  infinity  of  a  O.  P.
when  r<l.
1— f**        a         at^
w  1— r        1— r       1— r


GEOMETRIC  PROGRESSION  BW

Since  r<  1,  r*^  can  be  made  less  than  any  assignable  quantity
by  taking  n  suflSciently  large.     Hence  8,^  can  be  made  to

differ  from  by  a  quantity  less   than  any  assignable

quantity,  i.e,  5^  =  •

IV'ote.  Since  r«  continually  increases  in  magnitude  as  n
increases,  if  r  is  numerically  greater  than  unity,  it  is  seen  that
when  r  >  1,  the  sum  of  an  infinite  number  of  terms  is  oo.
265.  Any  sum  paid  at  regular  intervals  is  called  an  Annuity.
Unless  otherwise  stated  it  is  supposed  that  the  annuity  is  paid
yearly.
