5Q2 PHILOSOPHICAL TRANSACTIONS. [ANNO 1784. 



>« li — c) . + — « — " e x e ^+i . e q U ate it to the given term, and there results 1x 



2; + 1 . 2^ + 3 

 (1 — e) = 1 and 3* — ae = 2, and consequently e = 4- and x = f., if the 



series can he summed. 



The same observation, viz. that if any factor in the denominator or irrational 

 quantity have no other correspondent to it ; for example, if the factor be z + g, 

 and there is no correspondent one x + g + n, where n is a whole number, then 

 its integral cannot be expressed by a finite algebraical function of z. In the 

 same manner may the sums be found, when the terms are exponentials of supe- 

 rior orders ; for the exponential, irrational, &c. quantities in the denominators 

 of the sums may be easily deduced from the preceding principles ; and thence, 

 by proceeding as is before taught, the sum required. The principles of all these 

 cases have been given in the Meditationes. 



8. Mr. James Bernoulli found summable serieses by assuming a series v, 

 whose terms at an infinite distance are infinitely little, and subtracting the series 

 diminished by any number (/) of terms from the series itself, &c. 



It is observed in the Meditationes, that if t (tw), t (m + n), t (m -f- n -f n), 

 t (m -\- n -\- n -f- re"), &c. be the terms at m, m + n, m -\- n + ri, m -\- n + 

 n -f- n", &c. distances from the first, and «t (m) -f 6t (m + ") + ct (m + n + 

 n) + (It (m -J- n + n + n") + &c. be the general term, it will be summable, 

 when a-\-b\-c-\-d-\- &c. = O; the sum of the series will be a{i(m) + t 

 (m -jr 1) + t (m + 2) + . . . +.T (m + re -f- re' + n" + &c. — 1)) + b{-r(m + 

 re) + t (m + n -f 1 ) + T (m + re •+• 2) + . . . + t (ret + re + n' + re* -f- &c. 



— 1)) + c(t(w -f- « + «') + T(m + re + re' -f l) + . . . (t(?k |n-f«' + re" + 

 &c. — l)) -f- &c. = h. If the sum a + b + c -\- d + &c. be not = O, and 

 the series t(?») + T(m -f- 1) -f T ( m + 2) + & c - > n infinitum be a converging 

 one = s, then will the sum of the resulting series be (a -\- b -\- c + d -{- &c.) 

 s — (b -f c -f r/ + &c.) . (t w . . . + T m +"-') — (c -f d + 8cc.) (t'"+" . . . 



T m + " + "'' 1 ) — ((/ + &C.) . ( T ™ + " + "' -j- . . , T ™+n + »' + n"-l) _ g cc _ 



8. 2. Let the series v consist of terms, which have only one factor in the de- 

 nominator, and its numerator = 1 ; that is, let the general term be 



and the series consequently - + -\ 1- &c. = v; from the 



rz + e ^ J e'r + e'2r+e' 



before-mentioned addition or subtraction there follows 



a b c „ *z m + $z m -< + v: m_4 + &c 



+ =~L ZTZ + .. . o. ^ . + &C. = 



;■: + e ' rz + r + e ' rz + 2r + c ' ' ~rz+e.rz + r+e. rz + 2r + e . &c. 



where m is not greater than the number (n) of factors in the denominator dimi- 

 nished by unity. From a, (3, y, &c. r and e being given, easily can be acquired 

 by simple equations, or known theorems, the required co-efficients a, b, c, &c. 

 If vi = n — 1, and x and a-\-b + c + d+ &c. = O, then the sum of the series 

 resulting will be Unite. 



