sums of several classes of infinite series. 273 
{LzrCj j-f Mx*c 1>2 -\- NxV, 3 + &c. j 
+ 0 {LrV 2)I + Mx*c 2>2 + N x 6 c 2j3 + Scc. j 
+ ° 2 { LrV 3,i+ MxV 3 ,2+ Nx#t 3, 3 + &c - } 
+ o 3 1 LxV 4>i + MxV 4j2 + NxV 4)3 + &c. ]• 
-{- &c. &c. 
The first line vanishes on account of the value of c l n , and 
since 0 is an infinitesimal, the second line will be larger than 
the sum of all the rest, provided the multiplicators of the 
powers of 0 are finite ; if therefore the series LrV 2 I + M x*c 2 2 
+ N*rV^ 3 + &c. is finite since it is multiplied by 0, we may 
neglect the whole of the above expression: our next step 
must be to determine whether the series 
La? 2 1 1 3 — 2 3 -j- 3 3 — &c. | 4- Mx 4 | i s — 2 5 -}- 3 5 — See. J + 
4- N<a? 6 1 1 7 — 2 7 -J- 3 7 — See. | + &c. 
is finite or infinite. It has been observed by Euler, that the 
following relations exist between the direct and reciprocal 
powers of the natural numbers. 
1 — 2 + 3 — 4 + &c. = + + p + &c. ]• 
1 3 — 2 3 + 3 3 — 4 3 + &c. = — 2 ^{i + + T + &c. j 
l 5 — 2 s -\- 3 5 — 4 5 -f* &c. = 4-2 { 1 + ~p + je + &c. | 
1 7 — 2 7 4- 3 7 — 4 7 + &C. = — 2^- 7 {i 4- -p+ £ + &c. j 
&c. &c. 
These latter being substituted for their equals, we have 
— + p + &c. ) + + 
+ 7 &c. - sNx '^4 + + 4 + &c. + See. 
5 I \ i° • 3 s 1 s° 
The series which now multiplies each term is in all cases 
mdcccxix. N n 
