Direct comparison test | EPFL Graph Search

In mathematics, the comparison test, sometimes called the direct comparison test to distinguish it from similar related tests (especially the limit comparison test), provides a way of deducing the convergence or divergence of an infinite series or an improper integral. In both cases, the test works by comparing the given series or integral to one whose convergence properties are known. In calculus, the comparison test for series typically consists of a pair of statements about infinite series with non-negative (real-valued) terms: If the infinite series converges and for all sufficiently large n (that is, for all for some fixed value N), then the infinite series also converges. If the infinite series diverges and for all sufficiently large n, then the infinite series also diverges. Note that the series having larger terms is sometimes said to dominate (or eventually dominate) the series with smaller terms. Alternatively, the test may be stated in terms of absolute convergence, in which case it also applies to series with complex terms: If the infinite series is absolutely convergent and for all sufficiently large n, then the infinite series is also absolutely convergent. If the infinite series is not absolutely convergent and for all sufficiently large n, then the infinite series is also not absolutely convergent. Note that in this last statement, the series could still be conditionally convergent; for real-valued series, this could happen if the an are not all nonnegative. The second pair of statements are equivalent to the first in the case of real-valued series because converges absolutely if and only if , a series with nonnegative terms, converges. The proofs of all the statements given above are similar. Here is a proof of the third statement. Let and be infinite series such that converges absolutely (thus converges), and without loss of generality assume that for all positive integers n. Consider the partial sums Since converges absolutely, for some real number T. For all n, is a nondecreasing sequence and is nonincreasing.

Ecological momentary assessment of emotional processing: An exploratory analysis comparing daily life and a psychotherapy analogue session

Mehdi Gholam, Hélène Beuchat, Loris Grandjean

Background: Emotional processing has been studied in psychotherapy as a state-dependent, sequential process of change. So far, no studies have applied this conceptualisation of emotional processing to the assessments of emotion in daily life. This is particularly important in the light of the pertinence of day-by-day fluctuations of emotions for understanding mental health and for monitoring the impact of prevention and psychotherapy programmes. This study examined the internal and ecological validity of a state-dependent conceptualisation of emotional processing in daily life, in comparison with an experiential-psychodynamic psychotherapy analogue session. Methods: In total, N = 42 university students participated in an experiential-psychodynamic session, completed symptom measures and responded to a one-week period of ecological momentary assessment (EMA) using a smartphone. Emotional processing in the session was assessed using the valid observer-rated measure Classification of Affective Meaning States (CAMS), and emotional responses in daily life were assessed using newly developed theory-consistent items self-rated via an interactive smartphone program. Results: Internal validity was generally satisfactory across the subscales used in EMA. Correspondence between EMA and in-session emotional processing was generally low, but specific relationships were found between self-rated fear, rejecting anger, hurt/grief or loneliness and the observer-rated productive emotions in the psychotherapy analogue session. Relationships between maladaptive emotional processing and intensity in symptoms were found. Conclusions: This is the first study to have examined the validity of a state-dependent conception of emotional processing in daily life, in direct comparison with a psychological session. We recommend using this assessment schedule to develop or complement integrative prevention or intervention programmes.

WILEY2021

Ecological momentary assessment of emotional processing: An exploratory analysis comparing daily life and a psychotherapy analogue session

Mehdi Gholam, Hélène Beuchat, Loris Grandjean

WILEY2021