Find the sum of the series `Sigma_(r=1)^(n) r x^(r-1)` using calculus .

Sum of the series sum_(r=1)^(n) (r^(2)+1)r! is

Evaluate sum_(r=1)^(n)rxxr!

The sum of the series sum_(r=0)^(10) .^(20)C_(r) , is 2^(19)+{(.^(20)C_(10))/2} .

sum_(r=1)^n(2r+1)=...... .

If sum_(r=1)^(n)T_(r)=(n(n+1)(n+2)(n+3))/(12) where T_(r) denotes the rth term of the series. Find lim_(nto oo) sum_(r=1)^(n)(1)/(T_(r)) .

Find the remainder when sum_(r=1)^(n)r! is divided by 15, if n ge5 .

If Sigma_( i = 1)^( 2n) sin^(-1) x_(i) = n pi , then find the value of Sigma_( i = 1)^( 2n) x_(i) .

Two resistance R_(1) and R_(2) are made of different material. The temperature coefficient of the material of R_(1) is alpha and of the material of R_(2) is -beta . Then resistance of the series combination of R_(1) and R_(2) will not change with temperature, if R_(1)//R_(2) will not change with temperature if R_(1)//R_(2) equals

Two resistors of resistances R_(1)=100pm3 ohm and R_(2)=200pm4 ohm are connected (a) series , (b) in parallel. Find the equivalent resistance of the (a) series combination, (b) parallel combination. Use for (a) the relation R = R_(1)+R_(2) and for (b) (1)/(R') = (1)/(R_(1))+(1)/(R_(2)) and (DeltaR')/(R'^(2)) = (DeltaR_(1))/(R_(1)^(2))+(DeltaR_(2))/(R_(2))

A derivable function f : R^(+) rarr R satisfies the condition f(x) - f(y) ge log((x)/(y)) + x - y, AA x, y in R^(+) . If g denotes the derivative of f, then the value of the sum sum_(n=1)^(100) g((1)/(n)) is