Gottfried Leibniz: The Calculus Controversy n de 8.4 Problems Hint: Since her 1. Assuming Leibniz 's series za- 1 1 elet (1 +x)"=le logi+ du 4 3 5 full du 38 prove that = (1 + x)" log(1 +x), for 1 + 9.11 JT the above limit is the value of the derivative the 8 5.7 (1 +x)" at u 0. (b) Next use the binomial series expansion of (1+x) to obtain mie 1 1 1 22-1 62-1 + 102- 1 + her s ones with 2. Given that P= n! denotes the number of permutations of n objects, establish the following identities from Leibniz 's Ars Combinatoria: (1/n- 1) log(1 +x) = lim x+ 2! ench (1/n- 1)(1/n-2) 2 P,- (n - 1)P-1 = P,+P-1 was ww 3! P2= P-1(P+1Pr). and eath. were rvive (1/n - 1)(1/n -2)(1/n - 3) ww 3. Verify Leibniz's famous identity, + 4! V6=1+-3+ 1--3, .She (i-) (c) From the fact lim -k, conclude born. il she n00 n which gives an imaginary decomposition of the real number 6. that itors, death itting er the on the 4. Obtain Mercator's logarithmic series log(1+x) x - .3 log(1+x) x 2 X + 3 4 4 7. Show that the binomial theorem, as stated by New in his letter to Oldenburg, is equivalent to the more for -1 < x 1, by first calculating by long division the series сcom- on the ion, a familiar form 1 rir- 1) = 1 -x+ x - x3+ . 1 + x (1+x =1+rxr + 2! and then integrating termwise between 0 and x. r(r - 1)- 2) + e time isible se the upiter lation, www.d 5. Prove that 3! where r is an arbitrary integral or fractional expon The necessary condition x< 1 for convergence not stated by Newton. log x = 2 x + + 1 - x 7 for 1 < x < 1, and hence 8. Use the binomial theorem to obtain the following series expansions. comet log 2 = 2 it-on 1 (1+x)-l=1 -x+ x2 - r2 +.. + (-1)'x" +... curate 6. Supply the details of the following derivation, due to Euler, of the infinite series expansion for log(l +x): 3 5 35 3 33 either (a) (1 x)2=1+2r+ 3r2+.. +(n + 1)x" +. ... er than e same ly (a) (b) Show t on be given by the limit + + + + + 1 -123 11 II
Addition Rule of Probability
It simply refers to the likelihood of an event taking place whenever the occurrence of an event is uncertain. The probability of a single event can be calculated by dividing the number of successful trials of that event by the total number of trials.
Expected Value
When a large number of trials are performed for any random variable ‘X’, the predicted result is most likely the mean of all the outcomes for the random variable and it is known as expected value also known as expectation. The expected value, also known as the expectation, is denoted by: E(X).
Probability Distributions
Understanding probability is necessary to know the probability distributions. In statistics, probability is how the uncertainty of an event is measured. This event can be anything. The most common examples include tossing a coin, rolling a die, or choosing a card. Each of these events has multiple possibilities. Every such possibility is measured with the help of probability. To be more precise, the probability is used for calculating the occurrence of events that may or may not happen. Probability does not give sure results. Unless the probability of any event is 1, the different outcomes may or may not happen in real life, regardless of how less or how more their probability is.
Basic Probability
The simple definition of probability it is a chance of the occurrence of an event. It is defined in numerical form and the probability value is between 0 to 1. The probability value 0 indicates that there is no chance of that event occurring and the probability value 1 indicates that the event will occur. Sum of the probability value must be 1. The probability value is never a negative number. If it happens, then recheck the calculation.
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