Question

summation of 2^n/n! with limit 1 to infinity

Answer 1

Answer:

Method

Consider&#xA0;&#x2211;n=0&#x221E;xn=11&#x2212;x,&#xA0;where&#xA0;|x|&lt;1." role="presentation" style="display: inline-table; font-style: normal; font-weight: normal; line-height: normal; font-size: 15px; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;">Consider ∑n=0∞xn=11−x, where |x|<1.Consider ∑n=0∞xn=11−x, where |x|<1.

Then differentiating both sides w.r.t. x, we have

&#x2211;n=1&#x221E;nxn&#x2212;1=1(1&#x2212;x)2" role="presentation" style="display: table-cell !important; font-style: normal; font-weight: normal; line-height: normal; font-size: 15px; text-indent: 0px; text-align: center; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; width: 10000em !important; position: relative;">∑n=1∞nxn−1=1(1−x)2∑n=1∞nxn−1=1(1−x)2

Putting x=12" role="presentation" style="display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 15px; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;">x=12x=12 gives

&#x2211;n=1&#x221E;n2n&#x2212;1=1(1&#x2212;12)2=4" role="presentation" style="display: table-cell !important; font-style: normal; font-weight: normal; line-height: normal; font-size: 15px; text-indent: 0px; text-align: center; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; width: 10000em !important; position: relative;">∑n=1∞n2n−1=1(1−12)2=4∑n=1∞n2n−1=1(1−12)2=4

12&#x2211;n=1&#x221E;n2n&#x2212;1=2" role="presentation" style="display: table-cell !important; font-style: normal; font-weight: normal; line-height: normal; font-size: 15px; text-indent: 0px; text-align: center; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; width: 10000em !important; position: relative;">12∑n=1∞n2n−1=212∑n=1∞n2n−1=2

&#x2211;n=1&#x221E;n2n=2" role="presentation" style="display: table-cell !important; font-style: normal; font-weight: normal; line-height: normal; font-size: 15px; text-indent: 0px; text-align: center; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; width: 10000em !important; position: relative;">