2 Pn – (n – 1)P , = P,+ P,-1 P2 = P-(Pn+1- Pn). Dand 3. Verify Leibniz's famous identity, V6 = V1+ v-3+ V1-V-3, %3D which gives an imaginary decomposition of the real number /6. 4 Obtain Mercator's...

Number 4

Extracted text: 2 Pn – (n – 1)P , = P,+ P,-1 P2 = P-(Pn+1- Pn). Dand 3. Verify Leibniz's famous identity, V6 = V1+ v-3+ V1-V-3, %3D which gives an imaginary decomposition of the real number /6. 4 Obtain Mercator's logarithmic series x² log(1+x) = x – x4 +.. 4 3 7. S for –1 < x="">< 1,="" by="" first="" calculating="" by="" long="" division="" the="" series="" 1="1" –="" x="" +="" x²="" –="" x³="" +="" .="" .,="" 1+x="" and="" then="" integrating="" termwise="" between="" 0="" and="" x.="" 5.="" prove="" that="" )="" x7="" b()-2(.="" 1+x="" log="2" |="" x="" +e="" %3d="" for="" –="" 1="">< x="">< 1,="" and="" hence="" 8.="" 1="" 1="" +="" 5="" 35="" 1="" 1="" log="" 2="2" %3d="" +.="" 6.="" supply="" the="" details="" of="" the="" following="" derivation,="" due="" to="" euler,="" of="" the="" infinite="" series="" expansion="" for="" log(1+x):="" 3="" 33="">