MTH 253 – Calculus (Other Topics)Chapter 10 Part I (sections 1-6) Review1. Given the first few terms of a sequence, determine a formula for the general term.2. Given a sequence, find its limit or determine that it is divergent.3. Determine whether a sequence is monotone decreasing, monotone increasing, or neither.a. Consider differences of successive terms.- If 10n na a+- <, then the sequence is decreasing.- If 10n na a+- >, then the sequence is increasing.b. Consider ratios of successive terms- If 11nnaa+<, then the sequence is decreasing.- If 11nnaa+>, then the sequence is increasing.c. Consider the derivative of the associated continuous function.- If [ ]0ndadn<, then the sequence is decreasing.- If [ ]0ndadn>, then the sequence is increasing.4. Determine if a monotone sequence is convergent or divergent. That is, consider limnna�� or just determine if there is a bound or not (lower bound for decreasing; upper bound for increasing).5. Find several partial sums of a series 1kka�=� and use the nth partial sum to determine if a series converges or diverges. Hint: Write out the terms before simplifying.1 1s a=2 1 2s a a= +3 1 2 3s a a a= + +ggg?ns =6. Evaluate the sum of a geometric series. 0 if 11diverges otherwisekkararr�=�<�=-����7. Tests For Convergence: Series w/ all terms non-negative (name, state, & use these). If a test “fails,” that means that the test tells you nothing about the series.a. Divergence TestIf lim 0kku���, then the series ku� diverges. Otherwise, the test fails.b. Integral TestIf ( )f x is an increasing and continuous function over [ , ]a � such that ( )ku f k= for all k a�, then ku� and ( )af x dx�� either both converge or both diverge.c. p-SeriesIf ku�=1pk�, then ku� converges if 1p > and diverges otherwise.d. Comparison TestIf ka� and kb� are such that k ka b�, then- If ka� diverges than so does kb� (because it is bigger).- If kb� converges than so does ka� (because it is smaller).- Otherwise, the test fails.e. Limit Comparison TestIf ka� and kb� are series where limkkkabr��=, then- If 0r > and finite, then either both series converge or both series diverge.- Otherwise, the test fails.f. Ratio TestIf ku�is a series where 1limkkkuur+��=, then- If 1r <, the series converges.- If 1r >, the series diverges.- If 1r =, the test fails.g. Root TestIf ku�is a series where limkkkur��=, then- If 1r <, the series converges.- If 1r >, the series diverges.- If 1r =, the test fails.8. Tests For Convergence: Series w/ negative & positive terms (name, state, & use these). If a test “fails,” that means that the test tells you nothing about the series.a. Geometric Series (see #6 above)b. Alternating Series TestAn alternating series ( 1)kka-� converges if the following two conditions are true. Otherwise the test fails.-1 2 3 a a a� � �ggg -lim 0kka��=c. Absolute Convergence TestIf ka� converges, then ka�converges. Otherwise the test fails.9. Types of Convergencea. Converge Absolutelyka� converges absolutely if ka� converges.b. Diverge Absolutelyka� diverges absolutely if ka� diverges.d. Converge Conditionallyka� is conditionally convergent if ka� converges but ka�